Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 87, pp. 1–22. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu FOURTH-ORDER DIFFERENTIAL OPERATORS WITH INTERIOR DEGENERACY AND GENERALIZED WENTZELL BOUNDARY CONDITIONS ALESSANDRO CAMASTA, GENNI FRAGNELLI Communicated by Jerome A. Goldstein Abstract. In this article we consider the fourth-order operators A1u := (au′′)′′ and A2u := au′′′′ in divergence and non divergence form, where a : [0, 1] → R+ degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the genera- tion property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic prob- lems. 1. Introduction In mathematical analysis the boundary conditions associated with a differential operator usually involve the function and its derivatives (including Dirichlet, Neu- mann and Robin conditions). In some cases, as in Markov process theory, it is natural to include boundary conditions involving the operator itself (see [26, 34] for a detailed exposition). for a detailed exposition. In particular, if A denotes an elliptic operator, the parabolic problem ∂u ∂t −Au = 0 in Ω ⊂ Rn, t ≥ 0, is said to be equipped with the Wentzell boundary condition if Au(t, x) = 0 for x ∈ ∂Ω and all t ≥ 0. In the literature a more general boundary condition which arises naturally in the context of the heat equation is the generalized Wentzell boundary condition (GWBC) αAu(x) + β ∂u ∂n (x) + γu(x) = 0, x ∈ ∂Ω, (1.1) where (α, β, γ) 6= (0, 0, 0). Note that (α, β, γ) can depend on x as well. Surpris- ingly, these boundary conditions arise naturally as part of the formulation of the problem and are incorporated in the derivation of the heat equation itself. Addi- tional motivation for the study of evolution equations with (GWBC) comes from their possible interpretation as evolution equations with dynamical boundary con- ditions (for a general view on the role of Wentzell boundary conditions we refer to 2020 Mathematics Subject Classification. 47D06, 35K65, 47B25, 47N20. Key words and phrases. Degenerate operators in divergence and non divergence form; generalized Wentzell boundary conditions; interior degeneracy. ©2022. This work is licensed under a CC BY 4.0 license. Submitted September 28, 2021. Published December 29, 2022. 1 2 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 [11]). It is worth to mention that, in the case of heat equations, (GWBC) allow to take into account the action and the effect of heat sources on the boundary (see [26]). For a systematic study of the derivations and physical interpretations of Wentzell boundary conditions we refer, e.g., to [26], which covers heat and wave equations. On the other hand, for beam equations, Cahn-Hilliard equations and related models one can see, e.g., [23, 24, 25] and the references therein. Because of their importance and physical interpretation, we mention briefly other mathe- matical problems and contexts in which these boundary conditions appear. For example, if Ω is a smooth bounded domain in Rn and B is a formally symmetric differential operator on L2(Ω) with domain D(B) ⊂ C∞0 (Ω), a classical problem is to find all self-adjoint extensions of B (or all self-adjoint restrictions of B∗). This problem is solved abstractly by von Neumann and has been worked out in detail in some concrete cases. For example, if Ω = (0, 1) and B = 1 i d dx , the self-adjoint ex- tensions of B are determined by boundary conditions and are parametrized by the unit circle in C. An analogous problem is to consider B = ∆ with D(B) = C∞0 (Ω). On the space C0(Ω), in which C∞0 (Ω) is dense, B generates a positive contraction semigroup and an interesting question is to understand which extensions of B on C(Ω) have this property. In his pioneering work [34] Wentzell showed that such extensions are characterized by (Wentzell) boundary conditions of the form (1.1), where (α(x), β(x), γ(x)) 6= (0, 0, 0) for all x ∈ ∂Ω with α > 0, β ≥ 0 and γ ≥ 0 (Wentzell’s work generalizes previous results by Feller in one space dimension, see, e.g., [18]). Furthermore, other multiple applications are possible. For example, in [32] the authors consider a Dirichlet problem that describes the basic diffusion of particles in a locally compact space, endowed with a Radon measure. More- over, in a very general setting, they present an abstract version of the Wentzell boundary conditions (see also [31]). In the context of classical and quantum field theory on asymptotically anti-de Sitter spacetimes (AdS) and other spacetimes with boundaries, [12] studies a massive scalar field in AdS in d+ 1 spacetime dimensions subject to (GWBC) and it is highlighted that they are dynamical boundary condi- tions invariant under the action of the isometry group of the AdS boundary. The treatment of Wentzell boundary conditions in the classical and quantum field theo- retic literature appears also in [3], where the classical mechanical system of a finite string with point masses subject to harmonic potentials in the extrema is solved. (GWBC) are also considered in [33] in (d + 1)-dimensional Minkowski spacetime with one or two timelike boundaries. In particular, in [33] the author shows that the Wentzell boundary conditions ensure that the short-distance singularities of the two-point function for the boundary field has the form expected of a field living in a d-dimensional spacetime, contrary to other boundary conditions, for which the two- point function inherits the short-singularity of the (d + 1)-dimensional bulk. This seems to be a very desired feature for holographic purposes. In Biology a significant amount of interest has been devoted to the analysis of mathematical models arising in structured population dynamics and a very important problem is the choice of suitable boundary conditions for a biologically plausible and mathematically sound model. In this field [6] introduces and analyzes a structured population model, with so called distributed recruitment term and (GWBC), describing the dynamics of a population infected with a certain type of bacteria. We refer to [15, 30] for a model of structured populations with generalized Wentzell-Robin boundary condi- tions. On the other hand, for more general operators one can see [6, 15, 29, 30]. EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 3 Finally, it is well known that degenerate parabolic equations are widely used as mathematical models in the applied sciences to describe the evolution in time of a given system. For this reason, in recent years an increasing interest has been devoted to the study of differential degenerate operators in divergence or in non divergence form. In particular, after the new directions opened in [16, 20], great attention is given to the operators A1u := (au′)′ , A2u := au′′ with general Wentzell boundary conditions. Actually, in this article we are interested in fourth order operators since many problems that are relevant for applications are described by these operators. Among these applications we can find dealloying (corrosion processes), population dynam- ics, bacterial films, thin film, tumor growth, clustering of mussels and so on (see [7, Introduction] for some detailed references). Moreover, operators of this type with suitable domains involving different boundary conditions arise in a natural way in several contexts as beam analysis and Euler-Bernoulli beam theory (see [8, Introduction] for some related references in this field). The novelty of this article is that we prove the generation property for the degenerate fourth order operators A1u := (au′′)′′ , A2u := au′′′′ equipped with (GWBC) (see the main theorems in Sections 3-4), obtaining the existence of solutions for the associated parabolic Cauchy problems. Here a ∈ C[0, 1] is such that there exists x0 ∈ (0, 1) such that a(x0) = 0 and a(x) > 0 if x 6= x0. As far as we know, this is the first paper that deals with this problem; thus we intend to fill this gap following the ideas of [17] and [20]. The paper is organized as follows: in Section 2 we prove some preliminary results that hold if a degenerates in a general point x0 ∈ [0, 1]. In Section 3 and in Section 4 we assume that the degeneracy point x0 belongs to (0, 1) and we prove that the operators Ai, i = 1, 2, equipped with (GWBC) are non negative and self- adjoint with dense domain, obtaining the well posedness for the associated Cauchy problems. It is worth noting that in this paper we deal with real function spaces, but the assertions can be easily extended to the complex case. Notation: C denotes universal positive constants which are allowed to vary from line to line; ′ denotes the derivative of a function depending on the real space variable x. 2. Preliminary results In this section we recall some suitable weighted spaces and preliminary results given in [8], that will be crucial for the rest of the paper. For simplicity, we distin- guish between the case of a weakly degenerate function and of a strongly degenerate one. 2.1. Weakly degenerate case. First of all, we give the following definition on the function a. 4 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 Definition 2.1. A function a ∈ C[0, 1] is said to be weakly degenerate if there exists x0 ∈ [0, 1] such that a(x0) = 0, a(x) > 0 for all x ∈ [0, 1] \ {x0} and 1 a ∈ L 1(0, 1). As an example of a weakly degenerate function we can take a(x) = |x − x0|K , with 0 < K < 1. To deal with the divergence case, for any weakly degenerate function a ∈ C[0, 1], we introduce the weighted space Hi a(0, 1) := { u ∈ Hi−1(0, 1) : u(i−1) is absolutely continuous in [0,1], √ au(i) ∈ L2(0, 1) } , (2.1) endowed with the norm ‖u‖2Hia(0,1) := i−1∑ j=0 ‖u(j)‖2L2(0,1) + ‖ √ au(i)‖2L2(0,1) ∀u ∈ Hi a(0, 1), (2.2) i = 1, 2; here H0(0, 1) := L2(0, 1) and u(0) = u. The following proposition holds. Proposition 2.2. For u ∈ Hi a(0, 1), i = 1, 2, let ‖u‖2i,a := ‖u‖2L2(0,1) + ‖ √ au(i)‖2L2(0,1). Then ‖u‖Hia(0,1) and ‖u‖i,a are equivalent. To prove this proposition, the next result is essential. Lemma 2.3 ([27, Theorem 7.37]). Let I = (a, b), with a, b ∈ R, a < b, and let 1 ≤ p, q, r ≤ +∞ be such that 1 2q + 1 2p ≥ 1 r . Let u ∈W 2,1 loc (I), then there exists c = c(p, q, r) > 0 such that ‖u′‖Lr(I) ≤ cl 1 r−1− 1 q ‖u‖Lq(I) + cl1− 1 p+ 1 r ‖u′′‖Lp(I) (2.3) for every 0 < l < L1(I), where L1(I) is the one-dimensional measure of I. Proof of Proposition 2.2. For i = 1 the statement is obvious. Now, take i = 2 and u ∈ H2 a(0, 1). Clearly, ‖u‖22,a ≤ ‖u‖2H2 a(0,1) . For the other estimate, it is sufficient to prove ‖u′‖2L2(0,1) ≤ C ( ‖u‖2L2(0,1) + ‖ √ au′′‖2L2(0,1) ) , (2.4) for a positive constant C. To this aim, observe that u ∈ W 2,1 loc (0, 1). Thus, by Lemma 2.3 with I = (0, 1), p = 1, q = 2 and r = 2, one has ‖u′‖L2(0,1) ≤ cl−1‖u‖L2(0,1) + cl1/2‖u′′‖L1(0,1). (2.5) Now, by Hölder’s inequality, ‖u′′‖L1(0,1) = ∫ 1 0 √ a|u′′|√ a dx ≤ (∫ 1 0 (a(u′′)2)(x)dx )1/2(∫ 1 0 1 a(x) dx )1/2 = ‖ √ au′′‖L2(0,1) ∥∥1 a ∥∥1/2 L1(0,1) . EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 5 Using this inequality in (2.5), one has ‖u′‖2L2(0,1) ≤ C ( ‖u‖2L2(0,1) + ‖ √ au′′‖2L2(0,1) ) for some suitable constant C > 0 and the thesis follows. � Moreover, observe that u ∈ H2 a(0, 1) implies u′ ∈ H1 a(0, 1). Using the space H2 a(0, 1), we define Zw(0, 1) := {u ∈ H2 a(0, 1) : au′′ ∈ H2(0, 1)}. (2.6) To deal with the non divergence case, in place of L2(0, 1) and Hi a(0, 1), we consider the spaces L2 1/a(0, 1) := { u ∈ L2(0, 1) : ∫ 1 0 u2 a dx < +∞ } , Hi 1/a(0, 1) := L2 1/a(0, 1) ∩Hi(0, 1), with the respective norms ‖u‖2L2 1/a (0,1) := ∫ 1 0 u2 a dx ∀u ∈ L2 1/a(0, 1), and ‖u‖2Hi 1/a (0,1) := ‖u‖2L2 1/a (0,1) + i∑ j=1 ‖u(j)‖2L2(0,1) ∀u ∈ Hi 1/a(0, 1), (2.7) for i = 1, 2. Observe that for all u ∈ Hi 1/a(0, 1), one can prove that ‖u‖2 Hi 1/a (0,1) is equivalent to ‖u‖2i,1/a := ‖u‖2L2 1/a (0,1) + ‖u(i)‖2L2(0,1). For i = 1 the equivalence is obvious; for i = 2 it follows by [5, Chapter VIII]. Indeed, for all u ∈ Hi(0, 1), one has that ‖u‖Hi(0,1) is equivalent to ‖u‖L2(0,1)+‖u(i)‖L2(0,1), i = 1, 2. However, ‖u‖2L2(0,1) = ∫ 1 0 u2(x) a(x) a(x)dx ≤ C ∫ 1 0 u2(x) a(x) dx = C‖u‖2L2 1/a (0,1) for all u ∈ L2 1/a(0, 1) and, in particular, for all u ∈ Hi 1/a(0, 1). Hence ‖u‖i,1/a is equivalent to ‖u‖Hi 1/a (0,1), for all u ∈ Hi 1/a(0, 1) and for i = 1, 2. Finally, we consider the space Ww(0, 1) := { u ∈ H2 1/a(0, 1) : au′′′′ ∈ L2 1/a(0, 1) } . (2.8) Clearly, it is a trivial fact that, if au′′′′ ∈ L2 1/a(0, 1) then u′′′′ ∈ L1(0, 1) (since 1 a ∈ L 1(0, 1)) and, by [7, Lemma 2.1], u ∈W 4,1(0, 1). Using the same considerations as in [7, Proposition 3.1], the spaces Hi 1/a(0, 1) and Hi(0, 1), i = 1, 2, coincide algebraically and the two norms are equivalent. Indeed, for all u ∈ Hi(0, 1) it is sufficient to prove that there exists a positive constant C such that ‖u‖2L2 1/a (0,1) ≤ C ∥∥1 a ∥∥ L1(0,1) ; but this follows immediately taking C := ‖u‖2C[0,1]. 6 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 Hence, if u ∈ Ww(0, 1), then u ∈ C3[0, 1], and, in particular, (au(k))(x0) = 0, for all k = 0, 1, 2, 3, being a(x0) = 0 and u(k) ∈ C[0, 1], for all k = 0, 1, 2, 3. In the above spaces the following Green formulas hold. Lemma 2.4 ([8, Lemmas 2.1 and 3.1]). If a is weakly degenerate, then (i) for all (u, v) ∈ Zw(0, 1)×H2 a(0, 1)∫ 1 0 (au′′)′′v dx = [(au′′)′v]x=1 x=0 − [au′′v′]x=1 x=0 + ∫ 1 0 au′′v′′dx; (2.9) (ii) for all (u, v) ∈ Ww(0, 1)×H2 1/a(0, 1)∫ 1 0 u′′′′v dx = [u′′′v]x=1 x=0 − [u′′v′]x=1 x=0 + ∫ 1 0 u′′v′′dx. (2.10) We underline that in [8], to prove the previous lemma, the requirement 1 a ∈ L1(0, 1) is not used; actually it is sufficient to require a ∈ C[0, 1]. 2.2. Strongly degenerate case. In this subsection we consider another notion of degeneracy: the strongly one. Definition 2.5. A function a ∈ C[0, 1] is called strongly degenerate if there exists x0 ∈ [0, 1] such that a(x0) = 0, a(x) > 0 for all x ∈ [0, 1] \ {x0}, and 1 a /∈ L 1(0, 1). As an example of a strongly degenerate function a we can take a(x) = |x − x0|K , with K ≥ 1. For any strongly degenerate function a let us introduce the corresponding weighted spaces Hi a(0, 1) := { u ∈ Hi−1(0, 1) : u(i−1) is locally absolutely continuous in [0, 1] \ {x0} and √ au(i) ∈ L2(0, 1) } , (2.11) equipped with the norms (2.2), i = 1, 2. Also in this case, with an additional assumption on a, the analogous of Proposition 2.2 holds. Hypothesis 2.6. Assume that there exists K ∈ [1, 2) such that the function x 7→ |x− x0|K/a is (1) non increasing on the left of x0 and non decreasing on the right of x0, if x0 ∈ (0, 1); (2) non decreasing on the right of 0, if x0 = 0; (3) non increasing on the left of 1, if x0 = 1. Observe that the previous assumption on a is not surprising because it was already used in other papers; moreover this assumption and the requirement K ≥ 1 are satisfied by the prototype that we have in mind a(x) = |x− x0|K , with K ≥ 1. More precisely, since in the previous hypothesis we require that K < 2, as prototype we consider a(x) = |x− x0|K , with K ∈ [1, 2). Proposition 2.7. Assume Hypothesis 2.6, and for u ∈ H2 a(0, 1) set ‖u‖22,a := ‖u‖2L2(0,1) + ‖ √ au′′‖2L2(0,1). Then ‖u‖H2 a(0,1) and ‖u‖2,a are equivalent. EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 7 Proof. Obviously, there exists a positive constant C such that for all u ∈ H2 a(0, 1), ‖u‖2,a ≤ C‖u‖H2 a(0,1) . Now, we will prove the other inequality. Assume, for simplicity, x0 = 0. As a first step we prove that there exists a positive constant C such that∫ 1 0 (v′(x))2dx ≤ C‖ √ av′′‖2L2(0,1), (2.12) for all v ∈ X := {v ∈ H2 a(0, 1) : ∃ y0 ∈ (0, 1) such that v′(y0) = 0}. Take x ∈ (0, 1] and v ∈ X ; then there exists y0 ∈ (0, 1) such that v′(y0) = 0 and v′(x) = v′(x)− v′(y0) = ∫ x y0 v′′(y) √ a√ a dy ≤ ‖ √ av′′‖L2(0,1) (∫ x y0 1 a(t) dt )1/2 . Hence ∫ 1 0 (v′(x))2dx ≤ ‖ √ av′′‖2L2(0,1) ∫ 1 0 (∫ x y0 1 a(t) dt ) dx. Thus, it is sufficient to estimate the integral in the right-hand side of the above inequality. To this aim, we split the integral as follows:∫ 1 0 (∫ x y0 1 a(t) dt ) dx = ∫ y0 0 (∫ x y0 1 a(t) dt ) dx+ ∫ 1 y0 (∫ x y0 1 a(t) dt ) dx. Using the assumption on a, one has∫ 1 y0 (∫ x y0 1 a(t) dt ) dx ≤ C ∫ 1 y0 (∫ 1 y0 1 tK dt ) dx ≤ C y1−K0 a(1)(K−1) , K 6= 1, C − log(y0) a(1) , K = 1, for a positive constant C. Now, we consider the term ∫ y0 0 ( ∫ x y0 1 a(t)dt ) dx. Using again the assumptions on a and the fact that the constant in Hypothesis 2.6 is strictly less than 2, one has∣∣∣ ∫ y0 0 (∫ x y0 1 a(t) dt ) dx ∣∣∣ = ∣∣∣− ∫ y0 0 (∫ y0 x 1 a(t) dt ) dx ∣∣∣ = ∫ y0 0 (∫ y0 x 1 a(t) dt ) dx = ∫ y0 0 ∫ t 0 1 a(t) dx dt = ∫ y0 0 t a(t) dt = ∫ y0 0 tK a(t)tK−1 dt ≤ C ∫ y0 0 1 tK−1 dt ≤ C 1 (2−K) y2−K0 . 8 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 Thus, ∫ 1 0 (∫ x y0 1 a(t) dt ) dx ≤ C and∫ 1 0 (v′(x))2dx ≤ C‖ √ av′′‖2L2(0,1), for a positive constant C. Now, we will prove the thesis for all u ∈ H2 a(0, 1). To this aim, consider u ∈ H2 a(0, 1) and let P be the subspace of polynomials of degree one. Then, we can find a polynomial p1 of degree one such that ‖u− p1‖L2(0,1) = min p∈P ‖u− p‖L2(0,1). Set v := u − p1; then v ∈ H2 a(0, 1), v has at least two zeros and its derivative vanishes at least once (see Lemma 2.8 below). Hence v ∈ X and, by (2.12), there exists a positive constant C such that ‖v′‖2L2(0,1) = ∫ 1 0 (v′(x))2dx ≤ C‖ √ av′′‖2L2(0,1). Hence ‖u′‖L2(0,1) ≤ ‖v′‖L2(0,1) + ‖p′1‖L2(0,1) ≤ C‖ √ av′′‖L2(0,1) + ‖p′1‖L2(0,1). (2.13) It remains to estimate ‖p′1‖L2(0,1). To this aim, observe that obviously there exists a positive constant C such that ‖p′1‖L2(0,1) ≤ C‖p1‖L2(0,1). (2.14) Moreover, ‖v‖L2(0,1) = ‖u− p1‖L2(0,1) ≤ ‖u‖L2(0,1) and ‖p1‖L2(0,1) ≤ ‖u− p1‖L2(0,1) + ‖u‖L2(0,1) ≤ 2‖u‖L2(0,1). (2.15) Hence, by (2.14) and (2.15), it follows that ‖p′1‖L2(0,1) ≤ C‖u‖L2(0,1), (2.16) for a positive constant C. By (2.13) and (2.16), we obtain ‖u′‖L2(0,1) ≤ C‖u‖2,a, (2.17) thus the statement follows. The proof in the case x0 6= 0 is similar, so we omit it. � Clearly, the analogous of Proposition 2.2 holds in the case i = 1 with a strongly degenerate without additional assumption on the function a itself. Lemma 2.8. Let X := L2(0, 1) ∩ C[0, 1] and let P be the subspace of polynomials of degree one. For all u ∈ H2 a(0, 1), if p1 ∈ P is such that ‖u− p1‖L2(0,1) = min p∈P ‖u− p‖L2(0,1), then the function v := u− p1 has at least two zeros. EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 9 Proof. Assume that v(x) := u(x) − p1(x) 6= 0 for all x ∈ [0, 1]. Without loss of generality, we can assume v(x) > 0. Clearly, v(x) ≥ min[0,1](u − p1) =: α > 0. Assume that p1(x) = mx+ q, m, q ∈ R. Then p1 + α ∈ P and ‖u− p1‖2L2(0,1) − ‖u− p1 − α‖ 2 L2(0,1) = α ∫ 1 0 (−α+ 2u− 2mx− 2q)dx. Recalling that u(x)−mx− q ≥ α and α > 0, we have ‖u− p1‖2L2(0,1) − ‖u− p1 − α‖ 2 L2(0,1) ≥ 0. But this is not possible since, by assumption, p1 is such that ‖u− p1‖L2(0,1) = min p∈P ‖u− p‖L2(0,1). Now, assume that there exists only a point y0 ∈ [0, 1] such that v(y0) = 0 and v(x) 6= 0 for all x 6= y0. For ε ∈ ( 0, 2 ∫ 1 0 v(x)dx ) , consider gε(x) = v(x) − ε. Then one can prove that ‖gε‖2L2(0,1) < ‖v‖ 2 L2(0,1). Indeed ‖gε‖2L2(0,1) = ∫ 1 0 (v2 + ε2 − 2εv)dx < ∫ 1 0 v2dx if and only if ε ( ε− 2 ∫ 1 0 v dx ) < 0 ⇔ ε < 2 ∫ 1 0 v dx. Again, we find p(x) := p1(x) + ε ∈ P , such that ‖u− p‖L2(0,1) ≤ ‖u− p1‖L2(0,1) and this is not possible. Thus, the proof is complete. � Also in the strongly degenerate case we consider the space Zw(0, 1) given in (2.6), where H2 a(0, 1) is the one defined in (2.11). To distinguish the two spaces, we use the notation Zs(0, 1) if a is strongly degenerate. Thus, if u ∈ Zs(0, 1), u′ is locally absolutely continuous in [0, 1] \ {x0} and not absolutely continuous in [0, 1] as for the weakly degenerate case; so equality (2.9) is not true a priori. For this reason in [8] we characterize the space Zs(0, 1). In particular, we introduce the space X := { u ∈ H1(0, 1) : u′ is locally absolutely continuous in [0, 1] \ {x0}, au, au′ ∈ H1(0, 1), au′′ ∈ H2(0, 1), √ au′′ ∈ L2(0, 1), (au(k))(x0) = 0, for all k = 0, 1, 2 } . Using the definition of X one can easily obtain the following property, see [8, Lemma 2.2]. Lemma 2.9. For all u ∈ X we have that (1) |a(x)u(x)| ≤ ‖(au)′‖L2(0,1) √ |x− x0|, (2) |a(x)u′(x)| ≤ ‖(au′)′‖L2(0,1) √ |x− x0|, (3) |a(x)u′′(x)| ≤ ‖(au′′)′‖L2(0,1) √ |x− x0| for all x ∈ [0, 1]. Thanks to the previous estimates and the fact that 1 a 6∈ L 1(0, 1), one can prove the following characterization. 10 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 Proposition 2.10 ([8, Proposition 2.1]). The spaces X and Zs(0, 1) coincide. For the non divergence case we consider the same spaces as for the weakly de- generate case but, to prove a formula similar to (2.10), we have to characterize the space H2 1/a(0, 1). Thus, we introduce Y := { u ∈ H2 1/a(0, 1) : u(x0) = (au′)(x0) = 0 } and, proceeding as in [9] and [21] (if x0 ∈ {0, 1}) or as in [7] (if x0 ∈ (0, 1)), one can prove the following result. Proposition 2.11. If Hypothesis 2.6 is satisfied, then H2 1/a(0, 1) = Y. Hence, if Hypothesis 2.6 is satisfied, we can rewrite the space Ws(0, 1) defined as in (2.8) in the following way Ws(0, 1) = { u ∈ H2 1/a(0, 1) : u(x0) = (au′)(x0) = 0 and au′′′′ ∈ L2 1/a(0, 1) } . As for the weakly degenerate case, one can prove the following Green formulas: Lemma 2.12 ([8, Lemmas 2.3, 3.2]). If a is strongly degenerate, then (1) equality (2.9) holds for all (u, v) ∈ Zs(0, 1)×H2 a(0, 1); (2) assume Hypothesis 2.6: • if x0 ∈ (0, 1), then for all (u, v) ∈ Ws(0, 1)×H2 1/a(0, 1), the equality∫ 1 0 u′′′′vdx = [u′′′v]x=1 x=0 − [u′′v′]x=1 x=0 + [u′′v′] x+ 0 x− 0 + ∫ 1 0 u′′v′′dx, holds. Here u′′(x+0 ) = limδ→0+ u ′′(x0 + δ), u′′(x−0 ) = limδ→0+ u ′′(x0 − δ) and v′(x+0 ) = v′(x−0 ) = v′(x0); • if x0 = 0, then for all (u, v) ∈ Ws(0, 1)×H2 1/a(0, 1)∫ 1 0 u′′′′v dx = u′′′(1)v(1)− [u′′v′]x=1 x=0 + ∫ 1 0 u′′v′′dx; • if x0 = 1, then for all (u, v) ∈ Ws(0, 1)×H2 1/a(0, 1)∫ 1 0 u′′′′v dx = −u′′′(0)v(0)− [u′′v′]x=1 x=0 + ∫ 1 0 u′′v′′dx. Actually Proposition 2.11 and Lemma 2.12.2 are proved in [8] under a weaker assumption (see [8, Hypothesis 3.1]; naturally, Hypothesis 2.6 implies this assump- tion). However, here we consider a stronger assumption since we have to apply Lemma 2.12.2 under this hypothesis. 3. Operators in divergence form with generalized Wentzell boundary conditions and interior degeneracy Let us fix βj , γj ∈ R such that βj > 0 and γj ≤ 0, j = 0, 1. Consider a weakly or a strongly degenerate function a and assume that the degeneracy point x0 is in the interior of the domain. Let f : [0, 1]→ R be a continuous function such that∫ 1 0 |f(x)|2dx+ a(0)|f(0)|2 β0 + a(1)|f(1)|2 β1 ∈ R EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 11 and define Xµ to be the completion of C[0, 1] with respect to the norm ‖ · ‖Xµ , where ‖f‖2Xµ = ∫ 1 0 |f(x)|2dx+ a(0)|f(0)|2 β0 + a(1)|f(1)|2 β1 . From [16] and [20], it follows that Xµ := L2([0, 1], dµ), where dµ := dx|(0,1) ⊕ adS β |{0,1}, dx denotes the Lebesgue measure on (0, 1), β = (β0, β1), and adS β denotes the natural Dirac measure dS on {0, 1} with weight a β . More precisely, Xµ is a Hilbert space with respect to the inner product given by 〈f, g〉Xµ = ∫ 1 0 f(x)g(x) dx+ a(0)f(0)g(0) β0 + a(1)f(1)g(1) β1 , where f, g ∈ Xµ are written as (fχ(0,1), (f(0), f(1))), (gχ(0,1), (g(0), g(1))). We recall that, as usual, χ(0,1) denotes the characteristic function of the interval (0, 1). Clearly, Hi(0, 1) ⊆ Xµ, i = 0, 1, 2, where we recall H0(0, 1) = L2(0, 1). In par- ticular, C[0, 1] ⊆ Xµ ⊆ L2(0, 1). Now, define the operator in divergence form A1u := (au′′)′′ equipped with the following generalized Wentzell boundary condi- tions A1u(j) + (−1)j+1 βj a(j) (au′′)′(j) + γju(j) = 0, j = 0, 1, (3.1) and the additional boundary conditions u′′(0) = u′′(1) = 0. (3.2) We distinguish between the weakly degenerate case and the strongly degenerate one. If a is weakly degenerate we define the weighted space: Zµ(0, 1) := {u ∈ H2 a(0, 1) : au′′ ∈ H2(0, 1) and (au′′)′′ ∈ Xµ}. Clearly Zµ(0, 1) ⊆ Zw(0, 1), and using the definition of the space H2 a(0, 1), Zµ(0, 1) can be rewritten as Zµ(0, 1) = { u ∈ H1(0, 1) : u′ is absolutely continuous in [0,1], √ au′′ ∈ L2(0, 1), au′′ ∈ H2(0, 1), (au′′)′′ ∈ Xµ } . Let us observe that for any (u, v) ∈ Zµ(0, 1) × H2 a(0, 1) the Green formula (2.9) holds. Now, let us define the domain of the operator A1 through the following subspace of Zµ ⊆ Xµ: Dw(A1) := { u ∈ Zµ(0, 1) : (3.1) and (3.2) hold } . Then we can prove the following theorem. Theorem 3.1. The operator A1 : Dw(A1)→ Xµ is non negative, self-adjoint with dense domain. Thus −A1 generates a contraction semigroup. Proof. First of all we prove that A1 is symmetric and non negative on Xµ. 12 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 A1 is symmetric: take u, v ∈ Dw(A1). Then (u, v) ∈ Zµ(0, 1) ×H2 a(0, 1) and (2.9) holds. Consequently 〈A1u, v〉Xµ = ∫ 1 0 (au′′)′′v dx+ a(0)(au′′)′′(0)v(0) β0 + a(1)(au′′)′′(1)v(1) β1 = [(au′′)′v]x=1 x=0 − [au′′v′]x=1 x=0 + ∫ 1 0 au′′v′′dx + a(0)v(0) β0 ( β0 a(0) (au′′)′(0)− γ0u(0) ) − a(1)v(1) β1 ( β1 a(1) (au′′)′(1) + γ1u(1) ) = v(1)(au′′)′(1)− v(0)(au′′)′(0) + ∫ 1 0 au′′v′′dx− γ0 β0 a(0)v(0)u(0) − γ1 β1 a(1)v(1)u(1) + v(0)(au′′)′(0)− v(1)(au′′)′(1) = ∫ 1 0 au′′v′′dx− γ0 β0 a(0)v(0)u(0)− γ1 β1 a(1)v(1)u(1) = 〈u,A1v〉Xµ . A1 is non negative: for any u ∈ Dw(A1), according to the above calculations, one has 〈A1u, u〉Xµ = ∫ 1 0 a|u′′|2dx− γ0 β0 a(0)|u(0)|2 − γ1 β1 a(1)|u(1)|2 ≥ 0. Now we prove that λI +A1 is surjective for sufficiently large λ ∈ R. λI +A1 is surjective: consider the space H2 a(0, 1) with the inner product (u, v)1 := 〈u, v〉Xµ + 〈 √ au′′, √ av′′〉L2(0,1) ∀u, v ∈ H2 a(0, 1), which induces the norm ‖u‖2◦ := ‖u‖2Xµ + ‖ √ au′′‖2L2(0,1) ∀u ∈ H2 a(0, 1). It is well known that for all u ∈ H1(0, 1) |u(x)| ≤ C‖u‖H1(0,1), ∀x ∈ [0, 1] (3.3) for a suitable positive constant C. Thus, thanks to (2.4) and (3.3), one can prove that the norm ‖ · ‖◦ is equivalent to ‖ · ‖H2 a(0,1) . Moreover, H2 a(0, 1) ↪→ Xµ ↪→ (H2 a(0, 1))∗, where (H2 a(0, 1))∗ is the dual space of H2 a(0, 1) with respect to Xµ. Let f ∈ Xµ and define F : H2 a(0, 1)→ R such that F (v) = ∫ 1 0 fv dx+ a(0)f(0)v(0) β0 + a(1)f(1)v(1) β1 ∀v ∈ H2 a(0, 1). From H2 a(0, 1) ↪→ Xµ, it follows that F ∈ (H2 a(0, 1))∗. Now, we define L(u, v) := λ ∫ 1 0 uv dx+ ∫ 1 0 au′′v′′dx+ (λ− γ1)a(1) β1 u(1)v(1)+ (λ− γ0)a(0) β0 u(0)v(0), EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 13 for all u, v ∈ H2 a(0, 1). Clearly L(u, v) is a continuous bilinear form. Moreover, it is coercive. Indeed, if we take λ > γi, for i = 0, 1, then L(u, u) = λ ∫ 1 0 u2dx+ ∫ 1 0 a(u′′)2dx+ (λ− γ1)a(1) β1 u2(1) + (λ− γ0)a(0) β0 u2(0) ≥ α‖u‖2◦, for all u ∈ H2 a(0, 1), where α := min{λ, 1, λ− γ0, λ− γ1}. As a consequence, by the Lax-Milgram Theorem, there exists a unique u ∈ H2 a(0, 1) such that L(u, v) = F (v) for any v ∈ H2 a(0, 1). Hence, setting for simplicity Ci := a(i) βi , i = 0, 1, the previous equality can be rewritten as∫ 1 0 au′′v′′dx+ (λ− γ0)C0u(0)v(0) + (λ− γ1)C1u(1)v(1) = ∫ 1 0 (f − λu)v dx+ C0f(0)v(0) + C1f(1)v(1). (3.4) In particular, (3.4) holds for all v ∈ C∞c (0, 1) ⊆ H2 a(0, 1). Hence∫ 1 0 au′′v′′dx = ∫ 1 0 (f − λu)v dx. This implies that (au′′)′′ = f−λu a.e. in (0, 1). Hence, using the fact that f−λu ∈ Xµ ⊆ L2(0, 1), one has that (au′′)′′ ∈ Xµ and au′′ ∈ H2(0, 1). Thus u ∈ Zµ(0, 1). Now, we come back to (3.4). Using (2.9), (3.4) becomes∫ 1 0 (au′′)′′v dx− [(au′′)′v]x=1 x=0 + [au′′v′]x=1 x=0 − γ0C0u(0)v(0)− γ1C1u(1)v(1) = ∫ 1 0 (f − λu)v dx+ C0(f − λu)(0)v(0) + C1(f − λu)(1)v(1), (3.5) for all v ∈ H2 a(0, 1). Since (au′′)′′ = f − λu a.e. in (0, 1), (3.5) becomes − [(au′′)′v]x=1 x=0 + [au′′v′]x=1 x=0 − γ0C0u(0)v(0)− γ1C1u(1)v(1) = C0(f − λu)(0)v(0) + C1(f − λu)(1)v(1), (3.6) for all v ∈ H2 a(0, 1). Recalling that a(0) 6= 0 and a(1) 6= 0, we have immediately that u′′(0) = u′′(1) = 0. Thus (3.6) becomes − [(au′′)′v](1) + [(au′′)′v](0)− γ0C0u(0)v(0)− γ1C1u(1)v(1) = C0(au′′)′′(0)v(0) + C1(au′′)′′(1)v(1) for all v ∈ H2 a(0, 1). Hence A1u(j) + (−1)j+1 βj a(j) (au′′)′(j) + γju(j) = 0, j = 0, 1. This implies that u ∈ Dw(A1). Thus λI +A1 is surjective for λ sufficiently large. By Theorems 5.3 and 5.5 (see the Appendix), we know that −A1 is self-adjoint with upper bound 0, has dense domain and (−A1, Dw(A1)) generates a contraction semigroup. � 14 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 By Theorem 3.1 (see the Appendix), the problem ut(t, x) +A1u(t, x) = h(t, x), (t, x) ∈ (0, T )× (0, 1), A1u(t, j) + (−1)j+1 βj a(j) (auxx)x(t, j) + γju(t, j) = 0, t ∈ (0, T ), j = 0, 1, uxx(t, 0) = uxx(t, 1) = 0, t ∈ (0, T ), u(0, x) = u0(x), x ∈ (0, 1), (3.7) is well posed in the sense of Theorem 3.3 below. As a first step, we recall the following concept. Definition 3.2. If u0 ∈ Xµ and h ∈ L2(0, T ;Xµ), a function u is said to be a weak solution of (3.7) if u ∈ C ( [0, T ];Xµ ) ∩ L2 ( 0, T ;H2 a(0, 1) ) and∫ 1 0 u(T, x)ϕ(T, x) dx− ∫ 1 0 u0(x)ϕ(0, x) dx− ∫ (0,T )×(0,1) u(t, x)ϕt(t, x) dx dt + a(1)u(T, 1)ϕ(T, 1) β1 − a(1)u0(1)ϕ(0, 1) β1 − a(1) β1 ∫ T 0 u(t, 1)ϕt(t, 1)dt + a(0)u(T, 0)ϕ(T, 0) β0 − a(0)u0(0)ϕ(0, 0) β0 − a(0) β0 ∫ T 0 u(t, 0)ϕt(t, 0)dt = − ∫ (0,T )×(0,1) a(x)uxx(t, x)ϕxx(t, x) dx dt− γ1 β1 ∫ T 0 a(1)u(t, 1)ϕ(t, 1)dt − γ0 β0 ∫ T 0 a(0)u(t, 0)ϕ(t, 0)dt+ ∫ (0,T )×(0,1) h(t, x)ϕ(t, x) dx dt + ∫ T 0 a(1)h(t, 1)ϕ(t, 1) β1 dt+ ∫ T 0 a(0)h(t, 0)ϕ(t, 0) β0 dt for all ϕ ∈ H1(0, T ;Xµ) ∩ L2(0, T ;H2 a(0, 1)). Theorem 3.3. For all h ∈ L2(0, T ;Xµ) and u0 ∈ Xµ, there exists a unique solution u ∈ C ( [0, T ];Xµ ) ∩ L2 ( 0, T ;H2 a(0, 1) ) of (3.7) such that sup t∈[0,T ] ‖u(t)‖2Xµ + ∫ T 0 ‖u(t)‖2H2 a(0,1) dt ≤ CT ( ‖u0‖2Xµ + ‖h‖2L2(0,T ;Xµ) ) (3.8) for some positive constant CT . Moreover, if h ∈W 1,1(0, T ;Xµ) and u0 ∈ Dw(A1), then u ∈ C1 ( [0, T ];Xµ ) ∩ C ( [0, T ];Dw(A1) ) . (3.9) Proof. The assertion concerning the assumption u0 ∈ Xµ and the regularity of the solution u when u0 ∈ Dw(A1) is a consequence of the results in [2], [28, Chapter 3, Section 4, Theorem 4.1 and Remark 4.3] and of [10, Lemma 4.1.5, Proposition 4.1.6 and Proposition 4.3.9], [4, Propositions 3.2 and 3.3]. We only need to prove (3.8). Let us fix u0 ∈ Dw(A1) and consider the corresponding weak solution u ∈ EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 15 C1 ( [0, T ];Xµ ) ∩ C ( [0, T ];Dw(A1) ) . Now, we multiply the equation of (3.7) by u considering the inner product in Xµ; thus 1 2 d dt ‖u(t)‖2Xµ + ‖ √ auxx(t)‖2L2(0,1) − γ0 β0 a(0)u2(t, 0)− γ1 β1 a(1)u2(t, 1) ≤ 1 2 ‖u(t)‖2Xµ + 1 2 ‖h(t)‖2Xµ . Hence we deduce that d dt ‖u(t)‖2Xµ ≤ d dt ‖u(t)‖2Xµ + 2‖ √ auxx(t)‖2Xµ ≤ ‖u(t)‖2Xµ + ‖h(t)‖2Xµ . (3.10) By Gronwall’s Lemma for every t ∈ [0, T ], we obtain ‖u(t)‖2Xµ ≤ e T ( ‖u0‖2Xµ + ‖h‖2L2(0,T ;Xµ) ) . Thus, there exists a positive constant C such that sup t∈[0,T ] ‖u(t)‖2Xµ ≤ C ( ‖u0‖2Xµ + ‖h‖2L2(0,T ;Xµ) ) . (3.11) Integrating the second inequality of (3.10) over (0, T ) and using (3.11), we have∫ T 0 ‖ √ auxx(t)‖2L2(0,1)dt ≤ C ( ‖u0‖2Xµ + ‖h‖2L2(0,T ;Xµ) ) , for a suitable positive constant C, and the conclusion follows. Clearly, (3.8) and (3.9) hold also if u0 ∈ H2 a(0, 1), since Dw(A1) is dense in H2 a(0, 1). � Now we assume that a is strongly degenerate and we consider the operator (A1, Ds(A1)). Here A1 is defined as in the weakly degenerate case and Ds(A1) is defined as Dw(A1) where we have to consider H2 a(0, 1) defined in (2.11) in place of H2 a(0, 1) defined in (2.1). In particular, Ds(A1) := { u ∈ Zs,µ(0, 1) : (3.1) and (3.2) hold } , where Zs,µ(0, 1) := { u ∈ H1(0, 1) : u′ is locally absolutely continuous in[0, 1] \ {x0} and √ au′′ ∈ L2(0, 1), au′′ ∈ H2(0, 1), (au′′)′′ ∈ Xµ } . Clearly, the Green formula given in Lemma 2.12 for the divergence case still holds. Now, we define X̃ := { u ∈ H1(0, 1) : u′ is locally absolutely continuous in [0, 1] \ {x0}, au, au′ ∈ H1(0, 1), au′′ ∈ H2(0, 1), √ au′′ ∈ L2(0, 1), (au′′)′′ ∈ Xµ, (au(k))(x0) = 0, for all k = 0, 1, 2 } . Analogously to Proposition 2.10 one has that X̃ = Zs,µ(0, 1). As for the weakly degenerate case, one has the next result which contains the generation property in the strongly degenerate context. Theorem 3.4. Assume Hypothesis 2.6. The operator A1 : Ds(A1) → Xµ is non negative, self-adjoint with dense domain. Thus −A1 generates a contraction semi- group. 16 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 The proof of this theorem is analogous to the one of Theorem 3.1, so we omit it. In any case we underline that, thanks to (2.17) and (3.3), the two norms ‖ · ‖◦ and ‖ · ‖H2 a(0,1) are equivalent. Thus we can prove that λI +A1 is surjective, for λ sufficiently large, and the analogous of Theorem 3.3 holds for (3.7) if a is strongly degenerate. 4. Operators in non divergence form with generalized Wentzell boundary conditions and interior degeneracy As in the previous section, let us fix βj , γj ∈ R such that βj > 0, γj ≤ 0, j = 0, 1. Consider a weakly degenerate function a, and assume that the degeneracy point x0 belongs to (0, 1). Let f : [0, 1]→ R be a continuous function such that∫ 1 0 |f(x)|2 a dx+ |f(0)|2 β0 + |f(1)|2 β1 ∈ R and define Yµ to be the completion of C[0, 1] with respect to the norm ‖ ·‖Yµ , where ‖f‖2Yµ = ∫ 1 0 |f(x)|2 a dx+ |f(0)|2 β0 + |f(1)|2 β1 . By [16] and [20], it follows that Yµ := L2 1/a([0, 1], dµ), where dµ := dx a |(0,1) ⊕ dS β |{0,1}. As before, dx denotes the Lebesgue measure on (0, 1), β = (β0, β1), and dS β denotes the natural Dirac measure dS on {0, 1} with weight 1/β. In this way Yµ becomes a Hilbert space with the inner product given by 〈f, g〉Yµ = ∫ 1 0 f(x)g(x) a dx+ f(0)g(0) β0 + f(1)g(1) β1 , in which f, g ∈ Yµ are written as in Section 3. Clearly, Hi 1/a(0, 1) ⊆ Yµ, i = 1, 2; in particular, C[0, 1] ⊆ Yµ ⊆ L2 1/a(0, 1). Now we introduce the operator in non divergence form A2u := au′′′′ equipped with the general Wentzell boundary conditions A2u(j) + (−1)j+1βju ′′′(j) + γju(j) = 0, j = 0, 1, (4.1) and the additional boundary conditions (3.2). Moreover, we consider the weighted space: Wµ(0, 1) := { u ∈ H2 1/a(0, 1) : au′′′′ ∈ Yµ } . Now, since au′′′′ ∈ Yµ implies au′′′′ ∈ L2 1/a(0, 1), the same considerations made before Lemma 2.4 hold; in particular, if u ∈ Wµ(0, 1), then (au(k))(x0) = 0 for k = 0, 1, 2, 3. Moreover, observe that, for any (u, v) ∈ Wµ(0, 1) × H2 1/a(0, 1), the Green formula (2.10) holds and Wµ(0, 1) ⊆ Ww(0, 1). Now, let us define the domain of the operator A2 through the following subspace of Wµ ⊆ Yµ: Dw(A2) := { u ∈ Wµ(0, 1) : (4.1) and (3.2) hold } . Hence, we can prove the following result that establishes the main properties of this operator. EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 17 Theorem 4.1. If a is weakly degenerate, then the operator (A2, Dw(A2)) is self- adjoint and non-negative on Yµ with dense domain. Thus −A2 generates a con- traction semigroup. Proof. First, we prove the symmetry and non-negativity of A2. A2 is symmetric: let u, v ∈ Dw(A2); then (u, v) ∈ Wµ(0, 1) × H2 1/a(0, 1) and, by (2.10), we have 〈A2u, v〉Yµ = ∫ 1 0 au′′′′v a dx+ a(0)u′′′′(0)v(0) β0 + a(1)u′′′′(1)v(1) β1 = [u′′′v]x=1 x=0 − [u′′v′]x=1 x=0 + ∫ 1 0 u′′v′′dx+ v(0) β0 (β0u ′′′(0)− γ0u(0)) − v(1) β1 ( β1u ′′′(1) + γ1u(1) ) = ∫ 1 0 u′′v′′dx− γ0 β0 v(0)u(0)− γ1 β1 v(1)u(1) = 〈u,A2v〉Yµ . A2 is non negative: for any u ∈ Dw(A2), according to the above calculations, one has 〈A2u, u〉Yµ = ∫ 1 0 |u′′|2dx− γ0 β0 |u(0)|2 − γ1 β1 |u(1)|2 ≥ 0. Finally, we prove that λI +A2 is surjective for sufficiently large λ ∈ R. λI +A2 is surjective: consider the space H2 1/a(0, 1) with the inner product (u, v)2 := 〈u, v〉Yµ + 〈u′′, v′′〉L2(0,1) ∀u, v ∈ H2 1/a(0, 1), which induces the norm ‖u‖24 := ‖u‖2Yµ + ‖u′′‖2L2(0,1) ∀u ∈ H2 1/a(0, 1). Thus, thanks to (3.3), the norm ‖ · ‖4 is equivalent to ‖ · ‖H2 1/a (0,1). Moreover, H2 1/a(0, 1) ↪→ Yµ ↪→ (H2 1/a(0, 1))∗, where (H2 1/a(0, 1))∗ is the dual space of H2 1/a(0, 1) with respect to Yµ. Let f ∈ Yµ and define F : H2 1/a(0, 1)→ R such that F (v) = ∫ 1 0 fv a dx+ f(0)v(0) β0 + f(1)v(1) β1 ∀v ∈ H2 1/a(0, 1). From H2 1/a(0, 1) ↪→ Yµ, it follows that F ∈ (H2 1/a(0, 1))∗. Now, we define M(u, v) := λ ∫ 1 0 uv a dx+ ∫ 1 0 u′′v′′dx+ (λ− γ1) β1 u(1)v(1) + (λ− γ0) β0 u(0)v(0), for all u, v ∈ H2 1/a(0, 1). Clearly M(u, v) is a continuous bilinear form. Moreover, it is coercive. Indeed, taking λ > γi, for i = 0, 1, and δ := min{λ, 1, λ− γ0, λ− γ1}, M(u, u) = λ ∫ 1 0 u2 a dx+ ∫ 1 0 (u′′)2dx+ (λ− γ1) β1 u2(1) + (λ− γ0) β0 u2(0) ≥ δ‖u‖24, 18 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 for all u ∈ H2 1/a(0, 1). As a consequence, by the Lax-Milgram Theorem, there exists a unique u ∈ H2 1/a(0, 1) such that M(u, v) = F (v) for any v ∈ H2 1/a(0, 1). Hence, the previous equality can be rewritten as∫ 1 0 u′′v′′dx+ (λ− γ0) β0 u(0)v(0) + (λ− γ1) β1 u(1)v(1) = ∫ 1 0 f − λu a v dx+ f(0)v(0) β0 + f(1)v(1) β1 . (4.2) In particular, (4.2) holds for all v ∈ C∞c (0, 1) ⊆ H2 1/a(0, 1). Hence∫ 1 0 u′′v′′dx = ∫ 1 0 f − λu a v dx. In other words, the second distributional derivative of u′′ is equal to f−λu a a.e. in (0, 1). Hence, using the fact that f −λu ∈ Yµ ⊆ L2 1/a(0, 1), one has that au′′′′ ∈ Yµ. Thus u ∈ Wµ(0, 1). Now, we come back to (4.2). Using (2.10), (4.2) becomes∫ 1 0 u′′′′v dx− [u′′′v]x=1 x=0 + [u′′v′]x=1 x=0 − γ0 β0 u(0)v(0)− γ1 β1 u(1)v(1) = ∫ 1 0 f − λu a v dx + (f − λu)(0)v(0) β0 + (f − λu)(1)v(1) β1 (4.3) for all v ∈ H2 1/a(0, 1). Thanks to the fact that au′′′′ = f − λu a.e. in (0, 1), (4.3) becomes − [u′′′v]x=1 x=0 + [u′′v′]x=1 x=0 − γ0 β0 u(0)v(0)− γ1 β1 u(1)v(1) = (f − λu)(0) β0 v(0) + (f − λu)(1) β1 v(1), (4.4) for all v ∈ H2 1/a(0, 1). We have immediately that u′′(0) = u′′(1) = 0. Thus (4.4) becomes −[u′′′v](1) + [u′′′v](0)− γ0 β0 u(0)v(0)− γ1 β1 u(1)v(1) = (au′′′′)(0) β0 v(0) + (au′′′′)(1) β1 v(1) for all v ∈ H2 1/a(0, 1). Hence u ∈ H2 1/a(0, 1), au′′′′ ∈ Yµ, u satisfies (3.2) and A2u(j) + (−1)j+1βju ′′′(j) + γju(j) = 0, j = 0, 1. This implies that u ∈ Dw(A2). In other words λI+A2 is surjective for λ sufficiently large. Thanks to Theorems 5.3 and 5.5 (see the Appendix), we know that −A2 is self- adjoint with upper bound 0, has dense domain and (−A2, Dw(A2)) generates a contraction semigroup. � EJDE-2022/87 FOURTH-ORDER DIFFERENTIAL OPERATORS 19 As a consequence of Theorem 4.1, one has that the problem ut(t, x) +A2u(t, x) = h(t, x), (t, x) ∈ (0, T )× (0, 1), A2u(t, j) + (−1)j+1βjuxxx(t, j) + γju(t, j) = 0, t ∈ (0, T ), j = 0, 1, uxx(t, 0) = uxx(t, 1) = 0, t ∈ (0, T ), u(0, x) = u0(x), x ∈ (0, 1), (4.5) is well posed in the following sense. Definition 4.2. If u0 ∈ Yµ and h ∈ L2(0, T ;Yµ), a function u is said to be a weak solution of (4.5) if u ∈ C ( [0, T ];Yµ ) ∩ L2 ( 0, T ;H2 1/a(0, 1) ) and∫ 1 0 u(T, x)ϕ(T, x) a(x) dx− ∫ 1 0 u0(x)ϕ(0, x) a(x) dx− ∫ (0,T )×(0,1) u(t, x)ϕt(t, x) a(x) dx dt + u(T, 1)ϕ(T, 1) β1 − u0(1)ϕ(0, 1) β1 − 1 β1 ∫ T 0 u(t, 1)ϕt(t, 1)dt + u(T, 0)ϕ(T, 0) β0 − u0(0)ϕ(0, 0) β0 − 1 β0 ∫ T 0 u(t, 0)ϕt(t, 0)dt = − ∫ (0,T )×(0,1) uxx(t, x)ϕxx(t, x) dx dt− γ1 β1 ∫ T 0 u(t, 1)ϕ(t, 1)dt − γ0 β0 ∫ T 0 u(t, 0)ϕ(t, 0)dt+ ∫ (0,T )×(0,1) h(t, x) ϕ(t, x) a(x) dx dt + ∫ T 0 h(t, 1)ϕ(t, 1) β1 dt+ ∫ T 0 h(t, 0)ϕ(t, 0) β0 dt for all ϕ ∈ H1(0, T ;Yµ) ∩ L2(0, T ;H2 1/a(0, 1)). In particular, the following well posedness theorem holds. Theorem 4.3. For all h ∈ L2(0, T ;Yµ) and u0 ∈ Yµ, there exists a unique solution u ∈ C ( [0, T ];Yµ ) ∩ L2 ( 0, T ;H2 1/a(0, 1) ) of (4.5) such that sup t∈[0,T ] ‖u(t)‖2Yµ + ∫ T 0 ‖u(t)‖2H2 1/a (0,1)dt ≤ CT ( ‖u0‖2Yµ + ‖h‖2L2(0,T ;L2 1/a (0,1)) ) for some positive constant CT . Moreover, if h ∈ W 1,1(0, T ;L2 1/a(0, 1)) and u0 ∈ Dw(A2), then u ∈ C1 ( [0, T ];Yµ ) ∩ C ( [0, T ];Dw(A2) ) . 5. Appendix In this last section we just give some important results needed for the above proofs. These results are well known, and we write them to make the paper self- contained. 20 A. CAMASTA, G. FRAGNELLI EJDE-2022/87 Proposition 5.1 ([14, Chapter 2.3, page 90]). A linear operator A on the real Hilbert space H is dissipative if and only if 〈Au, u〉H ≤ 0 ∀u ∈ D(A). Definition 5.2. A linear operator A is bounded above in the Hilbert space H if there exists ω ∈ R such that 〈Au, u〉H ≤ ω〈u, u〉H for all u ∈ D(A). In this case ω is called an upper bound of A. Thus, by the previous proposition, we have that if A is dissipative on the real Hilbert space H, then it is bounded above with upper bound 0. Theorem 5.3 ([1, Theorem B.14]). Let A be a linear operator on the Hilbert space H and let ω ∈ R. The following assertions are equivalent: (1) A is self-adjoint with upper bound ω; (2) (a) 〈Au, v〉H = 〈v,Au〉H for all u, v ∈ D(A), (b) 〈Au, u〉H ≤ ω〈u, u〉H for all u ∈ D(A), (c) there exists λ > ω such that (λI −A) is surjective in H. Finally, we recall the following generation results. Theorem 5.4 ([14, Chapter 2.3, page 91]). A self-adjoint operator (A,D(A)) on a Hilbert space H generates a strongly continuous semigroup (of self-adjoint oper- ators) if and only if it is bounded above. Theorem 5.5 ([14, Corollary 3.20]). Let (A,D(A)) be a dissipative operator on a reflexive Banach space such that λI − A is surjective for some λ > 0. Then A is densely defined and generates a contraction semigroup. Acknowledgments. A. Camasta is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazio- nale di Alta Matematica (INdAM) and a member of UMI “Modellistica Socio- Epidemiologica (MSE)”. A. 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Richard; Time asymptotics of structured populations with dif- fusion and dynamic boundary conditions, Discrete Continuous Dyn. Syst. Ser. B, Vol. 23 (2018), 4087–4116. [31] D. Mugnolo, S. Romanelli; Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differ. Equ., Vol. 118 (2006), 1–20. [32] H. Vogt,J. Voigt; Wentzell boundary conditions in the context of Dirichlet forms, Adv. Differ. Equ., Vol. 8 (2003), 821–842. [33] J. Zahn; Generalized Wentzell boundary conditions and quantum field theory, Ann. Henri Poincaré, Vol. 19 (2018), 163–187. [34] A. D. Wentzell; On boundary conditions for multidimensional diffusion processes, Theory Probab. Appl., Vol. 4 (1959), 164–177. Alessandro Camasta Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy Email address: alessandro.camasta@uniba.it Genni Fragnelli Department of Ecology and Biology, Tuscia University, Largo dell’Università, 01100 Viterbo, Italy Email address: genni.fragnelli@unitus.it 1. Introduction 2. Preliminary results 2.1. Weakly degenerate case 2.2. Strongly degenerate case 3. Operators in divergence form with generalized Wentzell boundary conditions and interior degeneracy 4. Operators in non divergence form with generalized Wentzell boundary conditions and interior degeneracy 5. Appendix Acknowledgments References