Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 82, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM FOR INFINITELY DEGENERATE ELLIPTIC OPERATORS USMAN HAFEEZ, THÉO LAVIER, LUCAS WILLIAMS, LYUDMILA KOROBENKO Abstract. We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subellip- tic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem. 1. Introduction Consider the Dirichlet problem with a divergence form (degenerate) elliptic op- erator ∇ ·A∇u = f in Ω, u|∂Ω = 0, (1.1) where A is nonnegative semidefinite and has bounded measurable coefficients, and Ω is a bounded domain in Rn with sufficiently smooth boundary. We are interested in establishing sharp conditions on the matrix A that guarantee existence of bounded weak solutions. More precisely, we are looking for a function u from the degenerate Sobolev space ( W 1,2 A ) 0 (Ω) satisfying∫ ∇u ·A∇ϕ = − ∫ fϕ for every test function ϕ ∈ C1 0 (Ω) (in which case we say that u is a weak solution of (1.1)), as well as the qualitative estimate ‖u‖L∞(Ω) ≤ C‖f‖X for some appropriate normed space X. The case when A is uniformly elliptic has been completely settled by Nash [10], Moser [9], and DeGiorgi [1], and is now considered a classical theory [4]. When the eigenvalues of the matrix A are allowed to vanish, i.e. the operator is degenerate elliptic, the theory is far from complete. There are generally two cases considered in the literature: finite vanishing with 2010 Mathematics Subject Classification. 35A01, 35B65, 35D30, 35H99, 35J25, 46E36. Key words and phrases. Elliptic equations; infinite degeneracy; rough coefficients; Dirichlet problem; solvability; global boundedness; Orlicz-Sobolev inequality. c©2021. This work is licensed under a CC BY 4.0 license. Submitted July 30, 2021. Published September 23, 2021. 1 2 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 rough coefficients, and infinite vanishing with smooth coefficients. In the case of finite vanishing, the first generalizations of the Moser-DeGiorgi theory are due to Fabes, Kenig, and Serapioni [2], and Franchi and Lanconelli [3]. The latter deals with the case when one of the eigenvalues of A is constant, while others may vanish to finite order. Franchi and Lanconelli’s big idea was to use the subunit metric space associated with the operator, and adapt the classical Moser iteration to that setting. Using this approach, Sawyer and Wheeden [11] built on the work of Franchi and Lanconelli, among others, to further investigate regularity questions for subelliptic operators with rough coefficients. In particular, they showed that the (2σ, 2) weak Sobolev inequality with σ > 1 in the subunit metric space( 1 |B| ∫ B |w|2σ ) 1 2σ ≤ Cr ( 1 |B| ∫ B |∇Aw|2 )1/2 + C ( 1 |B| ∫ B |w|2 )1/2 (1.2) for all w ∈W 1,2 0 (B), is sufficient for solvability of the Dirichlet problem (1.1) when Ω = B, a subunit metric ball, with the quantitative estimate ‖u‖L∞(B) ≤ C‖f‖Lq(B) , where q > σ′, and σ′ is the dual of σ. Moreover, if the above estimate holds for q = σ′ then Sobolev inequality (1.2) holds (almost necessity). In this paper we investigate the same question for the case of the infinitely degenerate operator L = ∇ · A∇. More precisely, we make use of an analogue of (1.2), considering the more general Orlicz spaces, Lφ, instead of the traditional Lebesgue spaces. By a (φ, 2) Orlicz-Sobolev inequality we mean ‖w‖Lφ(B,dµ) ≤ C(r) (∫ B |∇Aw|2dµ )1/2 (1.3) for all w ∈ ( W 1,2 A ) 0 (B) and some Young function φ (typically satisfying φ(t) > t2 for all t > 1), see Section 2 for precise definitions, and the measure dµ = dx/|B|. There are a few recent results indicating that Orlicz-Sobolev inequalities of the type (1.3) are the correct substitute for (1.2) when the operator is infinitely degenerate. First, as has been shown in [5], a classical weak Sobolev inequality (1.2) implies the doubling property of the underlying metric measure space, and hence the degeneracy must be of finite type. On the other hand, in [6, 7] an abstract regularity theory for degenerate operators has been developed under the assumption of appropriate Orlicz-Sobolev inequalities (stronger versions of (1.3)). Moreover, for particular classes of infinitely degenerate operators these inequalities were proved to hold in the degenerate Sobolev spaces associated with the operator. In this paper we investigate the connection between the Dirichlet problem (1.1) and the validity of (1.3). In particular, we prove sufficiency and almost necessity of an Orlicz-Sobolev type inequality for the existence, uniqueness, and boundedness of weak solutions to infinitely degenerate elliptic partial differential equations with homogeneous Dirichlet boundary conditions. Our main results are as follows Theorem 1.1. Let L = ∇ · A∇ with bounded measurable non-negative semidef- inite matrix A, and d a metric on Rn. Suppose also that (1.3) holds for all w ∈ ( W 1,2 A ) 0 (B) and the metric ball B = Ω ⊂ Rn with φ satisfying φ(t) ≥ t2 for all t ≥ 0, and φ(t) ≥ t2(ln t)N , N > 1, for all t ≥ 1. If f ∈ L∞(B), then there exists a unique weak solution u ∈ ( W 1,2 A ) 0 (B) of (1.1) in the ball Ω = B and it EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 3 satisfies ‖u‖L∞(B) ≤ C‖f‖L∞(B). Theorem 1.2. Let ϕ be a Young function with ϕ̃ being its dual, and define φ by φ(t) = ϕ(t2) for all t ∈ R. Suppose that for every f ∈ Lϕ̃(B) there exists a unique weak solution u ∈ ( W 1,2 A ) 0 (B) of (1.1) in the ball Ω = B which satisfies ‖u‖L∞(B,dµ) ≤ C‖f‖Lϕ̃(B,dµ) , with dµ = dx |B| . Then Orlicz-Sobolev inequality (1.3) holds for all w ∈ ( W 1,2 A ) 0 (B). Remark 1.3. Note that in the above theorems we do not assume that the metric d is the subunit metric associated with A. In practice, to prove Orlicz-Sobolev inequality (1.3) one would need to work in a subunit metric space [3], or a measure space associated with the operator [2]. A version of the result in Theorem 1.2 and a sketch of the proof appears in [6, Sections 1 and 2 of Chapter 9]. It can be seen as a generalization of the subelliptic result [11, Lemma 102] with Lϕ replacing Lσ and Lφ replacing L2σ. In the subellip- tic case, the requirement on the right hand side is f ∈ Lq with q > σ′. In Theorem 1.2 we require f ∈ L∞, a strengthening of Lϕ̃. Note that just like in the subelliptic case, there is a gap between necessary and sufficient conditions. We suspect that the sufficient condition in Theorem 1.1 can be sharpened, but not with our current method of proof. At this point we do not know if the gap can be closed completely. The article is organized as follows. After giving some background and preliminar- ies in Section 2, we prove the existence and global boundedness of weak solutions, Theorem 1.1, in Section 3. Section 4 is devoted to the proof of Theorem 1.2, the necessity of Orlicz-Sobolev for solvability of the Dirichlet problem with a quantita- tive bound. The proof follows closely the proof of Lemma 102 in [11], and it also appears in [6, Sections 1 and 2 of Chapter 9 ]. However, the case of Orlicz-Sobolev spaces is more delicate, so we fill in the gaps and provide all the details. Finally, Section 5 provides some counterexamples demonstrating that the requirement on the right hand side in Theorem 1.1 cannot be significantly relaxed. More precisely, we give examples of equations admitting unbounded weak solutions in the case of Laplacian, subelliptic, and infinitely degenerate elliptic operators. 2. Preliminaries 2.1. Subunit metric spaces. We start this section with some background mate- rial on subunit metric spaces associated with degenerate operators, all of which can be found in [7, Chapter 7]. As mentioned in the Introduction, we do not assume the underlying metric space is the subunit metric space, however, it will be used to construct counterexamples in Section 5. 2.1.1. Degenerate Sobolev spaces. Let A be a nonnegative semidefinite bounded measurable matrix, and assume that A(x) ≈ B(x)trB(x), i.e., there exist positive constants c1 and c2 such that for a.e. x ∈ Ω and all ξ ∈ Rn, c1|B(x)ξ|2 ≤ ξ ·A(x)ξ ≤ c2|B(x)ξ|2, where B(x) is a Lipschitz continuous n × n real-valued matrix defined for x ∈ Ω. We define the A-gradient by ∇A = B(x)∇ , (2.1) 4 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 and the associated degenerate Sobolev space W 1,2 A (Ω) to have norm ‖v‖W 1,2 A ≡ (∫ Ω ( |v|2 +∇v ·A∇v ))1/2 = (∫ Ω ( |v|2 + |∇Av|2 ))1/2 . The space ( W 1,2 A ) 0 (Ω) is defined as the closure in W 1,2 A (Ω) of the subspace of Lipschitz continuous functions with compact support in Ω. Note that even though the definition of the A-gradient depends on the choice of the matrix B, all these definitions are equivalent, and associated Sobolev spaces are the same. Definition 2.1. Given u, v ∈W 1,2 A , define the inner product on the gradients of u and v to be 〈∇u,∇v〉 := ∇u ·A∇v = ∇utrA∇v. Furthermore, define the A semi-norm of ∇u to be [∇u]2A := 〈∇u,∇u〉. 2.1.2. Subunit metrics. We now define subunit (or control, or Carnot-Carathéodory) metric associated with the operator L = ∇ ·A∇, see [3]. Definition 2.2. A subunit curve is a Lipschitz curve γ : [0, r]→ Ω such that (γ′(t)ξ)2 ≤ ξ′A(γ(t))ξ, a.e. t ∈ [0, r], ∀ξ ∈ Rn. A subunit metric is defined by d(x, y) = inf{r > 0 : γ(0) = x, γ(r) = y, γ is a subunit in Ω}, and the subunit ball centered at x with radius r is B(x, r) = {y ∈ Ω : d(x, y) < r}. Franchi and Lanconelli [3] were the first to realize that the classical Moser itera- tion scheme can be adapted to certain degenerate operators (with one fixed constant eigenvalue) provided the Euclidean Rn is replaced by the subunit metric space. 2.2. Orlicz spaces. As mentioned in the introduction, we will work with Orlicz spaces, which can be seen as generalizations of Lebesgue spaces: power functions used do define Lebesgue spaces are replaced by more general Young functions. The material below is taken from [8]. Definition 2.3 ([8]). A function θ : R→ [0,∞] is a Young function if (1) θ is a convex, lower semicontinuous, [0,∞]-valued function on R. (2) θ is even and θ(0) = 0. (3) θ is non-trivial, i.e. it is different from the constant function θ(s) = 0 for s ∈ R. Note that Properties (1) and (2) imply that any Young function is non-decreasing on [0,∞). Definition 2.4 ([8]). Given a Young function, θ, the convex conjugate of θ, is defined as θ̃ = sup s∈R {st− θ(s)} ∈ [0,∞] for t ∈ R. We next define the Luxembourg norm, which in turn leads to the definition of an Orlicz space. EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 5 Definition 2.5. Let θ be a Young function, and Ω be a space with a σ-field and a σ-finite positive measure µ. For any measurable function on Ω we define the Luxembourg norm as ‖f‖Lθ = ‖f‖Lθ(Ω) := inf { k > 0 : ∫ Ω θ(f/k)dµ ≤ 1 } , (2.2) where inf(∅) = +∞. For a Young function θ, the associated Orlicz space is Lθ(Ω) = {f measurable : ‖f‖Lθ <∞}. The following proposition follows directly from (2.2). Proposition 2.6. Let θ1 and θ2 be two Young functions such that θ1(t) ≤ θ2(t) for all t ≥ 0. Then Lθ2 ⊆ Lθ1 , in particular, for every f ∈ Lθ2 it holds ‖f‖Lθ1 ≤ ‖f‖Lθ2 . An equivalent norm on Lθ given below is based on duality and will be used in some of the proofs contained in this paper. Definition 2.7. The Orlicz norm of a measurable function f is defined as |f |Lθ : = sup {∫ Ω fg dµ : g ∈ Lθ̃ and ‖g‖Lθ̃ ≤ 1 } = sup {∫ Ω fg dµ : g ∈ Lθ̃ and ∫ Ω θ̃(g)dµ ≤ 1 } . The Orclicz norm and the Luxembourg norm are equivalent, more precisely, ‖f‖Lθ ≤ |f |Lθ ≤ 2‖f‖Lθ . (2.3) Proposition 2.8 (Hölder Inequality [8]). Given a Young function θ, for any f ∈ Lθ(Ω) and g ∈ Lθ̃(Ω) it holds ∫ |fg|dµ ≤ 2‖f‖θ‖g‖θ̃. (2.4) In particular, fg ∈ L1. Finally, we define a particular family of Orlicz functions first introduced in [7] and employed in the adaptation of DeGiorgi iteration in the proof of Theorem 1.1 Definition 2.9. The family of Orlicz bump functions {ΦN}N>1 is given by ΦN (t) = { t(ln t)N , if t ≥ E = EN = e2N ; (lnE)N t, if 0 ≤ t ≤ E = EN = e2N . 3. Sufficiency This section is devoted to the proof of Theorem 1.1. First we show existence and uniqueness of weak solutions and then establish the quantitative boundedness estimate. 6 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 3.1. Existence of a unique weak solution. The proof is based on the Lax- Milgram theorem applied to the bilinear form B[u, v] defined on ( W 1,2 A ) 0 × ( W 1,2 A ) 0 . Proposition 3.1. Let Ω ⊂ Rn be a bounded subset, and A a nonnegative semi- definite n× n matrix with bounded measurable coefficients. Suppose that for every w ∈ ( WA 1,2 ) 0 (Ω) the following (2, 2) Sobolev inequality holds∫ Ω |w|2dx ≤ C(Ω) ∫ Ω |∇Aw|2dx. (3.1) Then the bilinear form B : ( WA 1,2 ) 0 (Ω)× ( WA 1,2 ) 0 (Ω)→ R defined by B[u, v] := ∫ Ω ∇u ·A∇v is bounded and coercive, i.e. (1) There exists α > 0 such that |B[u, v]| ≤ α‖u‖W 1,2 A ‖v‖W 1,2 A for all u, v ∈( WA 1,2 ) 0 (Ω). (2) There exists β > 0 such that β‖u‖2 W 1,2 A ≤ B[u, u] for all u ∈ ( WA 1,2 ) 0 (Ω). Proof. We begin by showing B is bounded. We have using Hölder’s inequality |B[u, v]| = ∣∣ ∫ ∇u ·A∇v∣∣ ≤ (∫ |∇u ·A∇u|)1/2(∫ |∇v ·A∇v| )1/2 ≤ (∫ u2 + ∫ |∇u ·A∇u| )1/2(∫ v2 + ∫ |∇v ·A∇v| )1/2 = ‖u‖W 1,2 A ‖v‖W 1,2 A for all u, v ∈ ( W 1,2 A ) 0 . To show the coercivity of the bilinear form B, condition (2), we use Sobolev inequality (3.1) to obtain B[u, u] = 1 2 B[u, u] + 1 2 B[u, u] = 1 2 ∫ Ω ∇u ·A∇u+ 1 2 B[u, u] = 1 2 ∫ Ω |∇Au|2 + 1 2 B[u, u] ≥ 1 2C ∫ Ω u2 + 1 2 B[u, u] = 1 2C ∫ Ω u2 + 1 2 ∫ Ω ∇u ·A∇u ≥ min { 1 2C , 1 2 }(∫ Ω u2 + ∫ Ω ∇u ·A∇u ) = β‖u‖2 W 1,2 A , where β = min{ 1 2C , 1 2} and C = C(Ω) from (3.1). Thus, the bilinear form B is bounded and coercive. � We are now ready to show the existence and uniqueness of the weak solution claimed in Theorem 1.1. This result in fact holds under a weaker assumption on the function f , we only need to require f ∈ L2(B). Theorem 3.2. Let L = ∇·A∇ with bounded measurable non-negative semidefinite matrix A, and d a metric on Rn, such that for any metric ball B = B(x, r) with 0 < r < ∞ it holds 0 < |B| < ∞. Suppose also that Sobolev inequality (3.1) holds EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 7 for all w ∈ ( W 1,2 A ) 0 (B) and some ball B = Ω ⊂ Rn. If f ∈ L2(B), then there exists a unique weak solution u ∈ ( W 1,2 A ) 0 (B) to the Dirichlet problem ∇ ·A∇u = f in B u|∂B = 0 . (3.2) Proof. Consider the linear functional (f, ·) : ( W 1,2 A ) 0 (B)→ R defined by (f, w) = − ∫ B fw, ∀w ∈ ( W 1,2 A ) 0 (B). Since f ∈ L2(B) and w ∈ ( W 1,2 A ) 0 (B) ⊂ L2(B) we have |(f, w)| ≤ C‖f‖L2(B)‖w‖W 1,2 A (B), which shows that this linear functional is bounded on ( W 1,2 A ) 0 (B). Therefore, by Proposition 3.1 and Lax-Milgram Theorem there exists a unique element, u ∈( W 1,2 A ) 0 (B), such that B[u,w] = (f, w) for all w ∈ ( W 1,2 A ) 0 (B). By definition of B[u,w] this means ∫ ∇u ·A∇w = − ∫ fw for all w ∈ ( W 1,2 A ) 0 (B), and we conclude that u is the unique weak solution to (3.2). � Corollary 3.3. Under the assumptions of Theorem 1.1 there exists a unique weak solution to (3.2). Proof. Suppose Orlicz-Sobolev inequality (1.3) holds with ϕ(t) ≥ t2 for all t ≥ 0. Moreover, the Orlicz space defined by ψ(t) = t2 coincides with L2. Therefore, by Proposition 2.6 the (2, 2) Sobolev inequality (3.1) holds and since f ∈ L∞(B) ⊂ L2(B), Theorem 3.2 applies. � 3.2. Global Boundedness of Weak Solutions. We now arrive at the proof of the global boundedness estimate for weak solutions. The proof closely follows the argument of [7, Chapter 4]. However, the Orlicz-Sobolev inequality we assume is weaker than the one in [7], while our assumption on the right hand side f is stronger. We therefore provide the details of the arguments that are necessary to verify in this new setting. We start with a Caccioppli inequality, which is an analogue of [7, Proposition 24 and Corollary 25]. Proposition 3.4. Let u be a weak solution to (1.1) on Ω = B and define u+ = max{u, 0}, then the following Caccioppoli inequality holds on the ball B∫ {x∈B:u(x)>0} |∇Au+|2 dµ ≤ ∫ {x∈B:u(x)>0} u+‖f‖L∞ dµ, where dµ = dx|B|. Proof. Let v = u+ then we have v ∈ ( W 1,2 A ) 0 (B) and therefore,∫ B ∇u ·A∇v dµ = − ∫ B fv dµ,∫ {x∈B:u(x)>0} ∇u ·A∇u+ dµ = − ∫ {x∈B:u(x)>0} fu+ dµ, 8 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82∫ {x∈B:u(x)>0} |∇Au+|2 dµ ≤ ∫ {x∈B:u(x)>0} u+‖f‖L∞ dµ. � Corollary 3.5. Let u be a weak solution to (1.1) in B, and suppose that for some P > 0 and a non-negative function v ∈W 1,2 A (B) it holds ‖f‖L∞ ≤ Pv(x), a.e. x ∈ {u > 0} ∩B. Then ‖∇Au+‖2L2 ≤ P ∫ (u+v) dµ. (3.3) Proof. By Proposition 3.4 we have∫ {u>0} |∇Au+|2 dµ ≤ ∫ {u>0} u+‖f‖L∞ dµ, and using the assumption gives∫ {u>0} |∇Au+|2 dµ ≤ ∫ {u>0} u+Pv dµ, ‖∇Au+‖2L2 ≤ P ∫ (u+v) dµ. � Lemma 3.6. Let ϕ be a Young function and let φ be defined by φ(t) = ϕ(t2). Then for all u ∈ Lφ(B), ‖u2‖Lϕ ≤ ‖u‖2Lφ ≤ 4‖u2‖Lϕ . Proof. For the first inequality using Definition 2.5 we need to show that∫ B ϕ ( u2 ‖u‖2 Lφ ) dµ ≤ 1. Using φ(t) = ϕ(t2) we have∫ B ϕ ( u2 ‖u‖2 Lφ ) dµ = ∫ B φ ( u ‖u‖Lφ ) dµ ≤ 1, which implies ‖u2‖Lϕ ≤ ‖u‖2Lφ . To show the second inequality we need to show that ∫ ϕ ( 4u2 ‖u‖2 Lφ ) dµ ≥ 1. Once again using φ(t) = ϕ(t2) we have∫ ϕ ( 4u2 ‖u‖2 Lφ ) = ∫ φ ( 2u ‖u‖Lφ ) . By definition (2.2), ‖u‖Lφ is the smallest number such that ∫ φ ( u ‖u‖ Lφ ) ≤ 1, and therefore ∫ ϕ ( 4u2 ‖w‖2 Lφ ) = ∫ ϕ ( u2 ‖u‖2 Lφ /4 ) = ∫ φ ( u ‖u‖Lφ/2 ) ≥ 1, which concludes the proof. � EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 9 We are now ready to prove the L∞ estimate in Theorem 1.1, and the argument follows closely the proof of [7, Proposition 27]. Theorem 3.7. Let L = ∇·A∇ with bounded measurable non-negative semidefinite matrix A, and d a metric on Rn, such that for any metric ball B = B(x, r) with 0 < r < ∞ it holds 0 < |B| < ∞. Suppose also that the following Orlicz-Sobolev inequality holds for all v ∈ ( W 1,2 A ) 0 (B) and the metric ball B ⊂ Rn, ‖v‖Lφ(B) ≤ C(B)‖∇Av‖L2(B), (3.4) where φ is defined by φ(t) = Φ(t2) with Φ = ΦN from Definition 2.9, for some N > 1. Then the unique weak solution u to (3.2) satisfies sup B |u| ≤ C‖f‖L∞(B). Proof. We first define the family of truncations uk = (u− Ck)+, where Ck = τ‖f‖L∞ ( 1− c(k + 1)−ε/2 ) , τ ≥ 1, and denote Uk ≡ ∫ B |uk|2 dµ, where dµ = dx/|B|. Since uk ∈ ( W 1,2 A ) 0 (B) for all k, using Hölder’s inequality for Orlicz spaces (2.4) we can write∫ u2 k+1 dµ ≤ C‖u2 k+1‖LΦ · ‖1‖ LΦ̃ {uk+1>0} , (3.5) where the norms are taken with respect to the measure µ. Our first goal is to bound the first factor on the right. Note that if uk+1 > 0 we have u > Ck+1 = τ‖f‖L∞ ( 1− c(k + 2)−ε/2 ) , which implies that uk = (u− Ck)+ > cτ‖f‖L∞ [ (k + 1)−ε/2 − (k + 2)−ε/2 ] = cτ‖f‖L∞(k + 1)−ε/2 [ 1− (k + 1 k + 2 )ε/2] ≥ cτ‖f‖L∞(k + 1)−ε/2 ( 1− k + 1 k + 2 ) ε 2 (k + 1 k + 2 ) ε 2−1 . Note that k+1 k+2 < 1 which allows us to conclude that uk ≥ ε 2 cτ‖f‖L∞(k + 2)−1− ε2 on the set where uk+1 > 0; thus ‖f‖L∞ ≤ 2 cτε (k + 2)1+ ε 2uk ≤ 2 cε (k + 2)1+ ε 2uk, (3.6) since τ ≥ 1. Next, since u is a weak solution it follows that u−Ck+1 is also a weak solution so (3.6) implies we can use (3.3) with v = uk and P = 2 cε (k+ 2)1+ ε 2 , which gives ∫ |∇Auk+1|2 dµ = ∫ ∣∣∇A(u− Ck+1 ) + ∣∣2 dµ ≤ C 2 cε (k + 2)1+ ε 2 ∫ (uk+1uk) dµ 10 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 ≤ C 2 cε (k + 2)1+ ε 2 ∫ u2 k dµ. Applying (3.4) and using Lemma 3.6 with ϕ = Φ we have ‖u2 k+1‖LΦ ≤ ‖uk+1‖2Lφ ≤ C‖∇Auk+1‖2, which combining with the above inequality gives ‖u2 k+1‖LΦ ≤ C(k + 2) 2+ε 2 ∫ u2 k dµ. (3.7) Now we want to bound the second factor on the right hand side of (3.5), ‖1‖LΦ̃ . Consider the function Γ(t) := 1 Φ̃−1( 1 t ) , and note that ∫ {uk+1>0} Φ̃ (1 a ) dµ = Φ̃ (1 a )µ({uk+1 > 0}big) for all a > 0. Now let a = Γ ( µ({uk+1 > 0}) ) = 1 Φ̃−1 ( 1 µ({uk+1>0}) ) , so that ∫ {uk+1>0} Φ̃ (1 a ) dµ = 1, and therefore ‖1‖ LΦ̃ ( {uk+1>0} ) ≤ a = Γ ( µ({uk+1 > 0}) ) . (3.8) Now recall that we showed {uk+1 > 0} ⊂ { uk > ε 2 cτ‖f‖L∞(k + 2)−1− ε2 } , where τ ≥ 1 which follows from the observation that uk+1 > 0 implies uk > τ‖f‖L∞ ( 1− c(k + 2)−ε/2 ) . Using Chebyshev’s inequality thus gives µ ({uk+1 > 0}) ≤ µ ({ uk > ε 2 cτ‖f‖L∞(k + 2)−1− ε2 }) ≤ 4 c2τ2‖f‖2L∞ε2 (k + 2)2+ε ∫ u2 k dµ. (3.9) Combining (3.8) and (3.9) we obtain ‖1‖ LΦ̃ {uk+1>0} ≤ Γ ( C(k + 2)2+ε ∫ u2 k ) . (3.10) Finally substituting (3.7) and (3.10) into (3.5) we conclude that∫ u2 k+1 dµ ≤ C(k + 2) 2+ε 2 ∫ u2 k · Γ ( C(k + 2)2+ε ∫ u2 k ) , Uk+1 ≤ C(k + 2) 2+ε 2 UkΓ ( C(k + 2)2+εUk ) . EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 11 This estimate is the same as the one obtained in the proof of [7, Theorem 30], so the rest of the proof can be repeated verbatim to conclude that sup B |u| ≤ C‖f‖L∞(B). � 4. Almost necessity In this section we demonstrate the almost necessity of an Orlicz-Sobolev inequal- ity for the existence, uniqueness, and boundedness of solutions to (1.1), namely we prove Theorem 1.2. We start with two simple technical lemmas. Lemma 4.1. Let ϕ : R → [0,∞] be a Young function and let ϕ̃ be the convex conjugate, or dual, of ϕ as defined in Definition 2.4. Let B ⊂ Rn be any ball, and define X = {f ∈ Lϕ̃ : ∫ B ϕ̃(|f |)dµ ≤ 1}, Y = {f ∈ Lϕ̃ : f ≥ 0 and ∫ B ϕ̃(f)dµ ≤ 1}. Then sup X ∫ B w2f dµ = sup Y ∫ B w2f dµ (4.1) for any w ∈ Lip0(B). Proof. First note that Y ⊂ X. Thus, it suffices to show that∫ B w2g ≤ sup Y ∫ B w2f for all g ∈ X\Y. Let g ∈ X\Y , and write g+ and g− for the positive and negative parts of g respec- tively. Note that since ϕ̃ is even and non-decreasing on [0,∞) and non-negative on R, we have ∫ B ϕ̃(g+) ≤ ∫ B ϕ̃(g+ + g−) = ∫ B ϕ̃(|g|) ≤ 1 for all g ∈ X. In particular, we conclude that g+ ∈ Y . Therefore,∫ B w2g = ∫ B w2g+ − ∫ B w2g− ≤ sup f∈Y ∫ B w2f − ∫ w2g− ≤ sup f∈Y ∫ B w2f. The last inequality follows from the fact that since w2 and g− are both non-negative, it must be the case that ∫ B w2g− is also non-negative. � Lemma 4.2. Let ϕ : R → [0,∞] be a Young function and ϕ̃ be its dual. Further- more, define the sets X and Y as in Lemma 4.1. Then for all w ∈ Lip0(B), ‖u‖Lϕ ≤ sup f∈Y ∫ B uf dµ. Proof. As before let dµ = dx/|B|, and recall from (2.3) that ‖u‖Lϕ ≤ |u|Lϕ = sup {∫ ug dµ : ∫ ϕ̃(g) ≤ 1 } = sup {∫ ug dµ : ∫ ϕ̃(|g|) ≤ 1 } , where the last equality holds because ϕ̃ is even by definition of a Young function. Finally, using Lemma 4.1 we have sup {∫ ug dµ : ∫ ϕ̃(|g|) ≤ 1 } = sup g∈X ∫ B ug dµ ≤ sup g∈Y ∫ B ug dµ, which concludes the result. � 12 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 We are now ready to prove Theorem 1.2, which we state again here for conve- nience. Theorem 4.3. Let ϕ be a Young function that satisfies ϕ(t) > t for all t > 0, and let φ(t) = ϕ(t2). Additionally, let f ∈ Lϕ̃(B) and assume that all weak solutions u ∈( W 1,2 A ) 0 (B) to (3.2) satisfy the global boundedness estimate supB |u| ≤ C‖f‖Lϕ̃(B). Then the following Orlicz-Sobolev inequality holds: ‖v‖Lφ(B,dµ) ≤ C‖∇Av‖L2(B,dµ) for all v ∈ ( W 1,2 A ) 0 (B). Note that the global boundedness condition is different from that in the suffi- ciency result. We previously demonstrated that a (φ, 2) Orlicz-Sobolev inequality with sufficiently large φ gives the estimate, supB |u| ≤ C‖f‖L∞(B) for all weak solutions u to (3.2) with f ∈ L∞(B). However, in order to prove necessity of a (φ, 2) Orlicz-Sobolev inequality we require a stronger condition; namely, that all weak solutions to (1.1) with the right hand side in a larger class, i.e. f ∈ Lϕ̃(B), are bounded. Hence the term “almost necessity”. Proof. The proof is similar to the proof of [11, Lemma 102], and the proof in [6, Sections 1 and 2 of Chapter 9]. Let u ∈ ( W 1,2 A ) 0 be a weak solution to (3.2), i.e.,∫ B ∇ψ ·A∇u dµ = − ∫ B ψf dµ for all ψ ∈ Lip0(B), and assume f ≥ 0. For any w ∈ Lip0(B) we therefore have − ∫ B ∇w2 ·A∇u = ∫ B w2f, since w2 ∈ Lip0(B). By applying the chain rule to ∇w2 and using the inner product from Definition 2.1, we see that∫ B w2f = −2 ∫ B w〈∇w,∇u〉 ≤ 2 (∫ B w2[∇u]2A )1/2(∫ B [∇w]2A )1/2 , (4.2) where the inequality follows from an application of the Cauchy-Schwartz inequality followed by the Hölder’s inequality. Now analyzing the first term in the above inequality we observe that∫ B w2[∇u]2A = ∫ B w2∇u ·A∇u. (4.3) Furthermore, since Lip0(B) is dense in ( W 1,2 A ) 0 we can take w2u as a test function in the definition of a weak solution to obtain − ∫ w2uf = ∫ ∇(w2u) ·A∇u = ∫ (2w∇wu+ w2∇u) ·A∇u = 2 ∫ wu∇w ·A∇u+ ∫ B w2∇u ·A∇u. Therefore, ∫ w2∇u ·A∇u = −2 ∫ wu∇w ·A∇u− ∫ w2uf. Hence, (4.3) becomes∫ w2[∇u]2A = ∫ w2∇u ·A∇u = −2 ∫ wu∇w ·A∇u− ∫ w2uf EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 13 = −2 ∫ 〈u∇w,w∇u〉 − ∫ uw2f ≤ 1 2 ∫ w2[∇u]2A + 8 ∫ u2[∇w]2A + ∫ |u|w2|f |, where the final estimate follows from the Cauchy-Schwartz Inequality followed by Young’s Inequality. Absorbing the first term on the right to the left-hand side results in ∫ w2[∇u]2A ≤ C ( sup B |u| )2 ∫ [∇w]2A + C ( sup B |u| ) ∫ w2|f | ≤ C max {( sup B |u| )2 ∫ [∇w]2A, ( sup B |u| ) ∫ w2|f | } = C max {( sup B |u| )2 ∫ [∇w]2A, ( sup B |u| ) ∫ w2f } , where the last equality follows from the assumption that f is non-negative. We claim that comparing ∫ w2[∇u]2A to either term inside the maximum results in equivalent inequalities. First assume that the first term, (supB |u|) 2 ∫ [∇w]2A, dom- inates. Then combining the above inequality with (4.2) gives∫ w2f ≤ C ( sup B |u| ) ∫ [∇w]2A = C ( sup B |u| ) ‖∇Aw‖2L2 ≤ C‖f‖Lϕ̃‖∇Aw‖2L2 , where the final inequality follows from the global boundedness estimate. On the other hand, if supB |u| ∫ w2f dominates, then (4.2) gives∫ w2f ≤ C ( sup B |u| ∫ w2f )1/2(∫ [∇w]2A )1/2 . Combining with the global boundedness estimate, this becomes∫ w2f ≤ ‖f‖Lϕ̃ ∫ [∇w]2A = C‖f‖Lϕ̃‖∇Aw‖2L2 , (4.4) which is the same estimate as above. Using the equivalent definition of the Orlicz norm (2.7) and Lemma 4.1, we have |w2|Lϕ = sup {∫ B w2f : ∫ B ϕ̃(f) ≤ 1 and f ≥ 0 } = sup {∫ B w2f : ‖f‖Lϕ̃ ≤ 1 and f ≥ 0 } . Combining with (4.4) and (2.3) gives ‖w2‖Lϕ ≤ C‖∇Aw‖2L2 . To obtain the desired (φ, 2) Orlicz-Sobolev Inequality it remains to show that ‖w‖2Lφ ≤ C‖w 2‖Lϕ , which follows immediately from the second inequality in Lemma 3.6. Hence, ‖w‖2Lφ ≤ 4‖w2‖Lϕ ≤ C‖∇Aw‖2L2 . By the density of Lip0(B) in W 1,2 0 (B) we obtain the desired Orlicz-Sobolev inequal- ity, ‖v‖Lφ(B,dµ) ≤ ‖∇Av‖L2(B,dµ) for all v ∈ ( W 1,2 A ) 0 � 14 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 5. Sharpness In this section, we demonstrate a weak degree of sharpness of our results. More precisely, we show that even though the requirement on the right hand side function f in the sufficiency result, Theorem 1.1, is stronger then the one in Theorem 1.2, it cannot be significantly relaxed. Namely, there exist an operator A and a function u ∈ ( W 1,2 A ) 0 such that (1) a (Ψ, 2) Orlicz-Sobolev inequality holds in a subunit metric ball B with Ψ(t) = t2(ln t)N , N > 1, for all t > 1; (2) Lu ∈ LΦ̃M with ΦM (t) = t(ln t)M , M > 2 + 2N , for all t > 1; (3) u is unbounded at the origin. To set the stage, we first provide similar constructions in the case of the Laplacian operator, and a finitely degenerate elliptic operator. 5.1. Laplacian counterexample. Recall that in general we are concerned with the following divergence form operator Lu = ∇ · A∇u. Now consider the two dimensional case of R2 and let A = ( 1 0 0 1 ) , so L = ∆, the Laplace operator. For a generic u changing to polar coordinates gives ∆u = 1 r ∂ ∂r ( r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 . Now we choose a weak solution u that is unbounded at the origin. The power α helps us control the integrability of this unbounded function. Define u = ( ln 1 r )α , where 0 < α < 1/2, so one can check that u ∈W 1,2(B(0, 1/2)). Since this function does not depend on θ, we have Lu = ∆u = 1 r2 α(α− 1) ln (1 r )α−2 . Thus with f := 1 r2 α(α− 1) ( ln 1 r )α−2 , u is a weak solution to Lu = f which is unbounded at the origin. We now calculate the Lq norm of f in the ball B = B(0, 1/2) ‖f‖Lq(B) = ∫ 2π 0 ∫ 1/2 0 |f(r)|qr dr dθ = 2πα(α− 1) ∫ 1/2 0 ∣∣ 1 r2 ( ln 1 r )α−2∣∣q r dr ≈ ∫ 1/2 0 ( ln 1 r )q(α−2) dr r2q−1 . The integral on the right is finite provided q < 1 or q = 1 and α < 1. In particular, u = ln(1/r)1/4 is an unbounded weak solution to ∆u = f with f ∈ Lq(B), q = 1 = n/2. On the other hand, if u is a weak solution to Lu = f and f ∈ Lq(B) with q > n/2, [4, Theorem 8.16] gives that sup B |u| ≤ sup ∂B |u|+ C‖f‖q <∞. EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 15 Furthermore, for the Laplace operator, the associated subunit metric space co- incides with the Euclidean Rn, and we have the following Sobolev Inequality( 1 |B| ∫ B |w|2σ ) 1 2σ ≤ Cr ( 1 |B| ∫ B |∇Aw|2 )1/2 + C ( 1 |B| ∫ B |w|2 )1/2 (5.1) for σ ≤ n n−2 and so σ′ = n/2 is the dual of σ. 5.2. Degenerate counterexamples. The following two examples are based on the examples constructed in [6, Section 3 Chapter 9]. Let L = ∇ ·A∇ with A = ( 1 0 0 g(x)2 ) , where g(0) = 0, g is positive away from the origin, and g = ψ′ where ψ is smooth, even, strictly convex on R and ψ(0) = 0. Moreover, we will assume that g(x)/x→ 0 and ψ(x)/x2 → 0 as x → 0. Since the operator L is elliptic away from the y- axis, and translation invariant with respect to the y variable, we may restrict our attention to the ball B = B(0, ρ) centered at the origin, with radius ρ sufficiently small. Define the function u by u(x, y) := χ ( y ψ(x) ) ln 1 x , (5.2) where χ(s) is a smooth odd function on R such that χ(s) = 1 for s ∈ [−1, 1] and χ(s) = 0 for s ∈ R r [−2, 2]. First note that the function u is supported in the narrow region along the x-axis, where |y| ≤ 2ψ(x). Next we calculate uy = χ′ ( y ψ(x) ) 1 ψ(x) ln 1 x , uyy = χ′′ ( y ψ(x) ) 1 ψ(x)2 ln 1 x , ux = χ′ ( y ψ(x) )(−yψ′(x) ψ(x)2 ) ln 1 x − 1 x χ ( y ψ(x) ) , uxx = χ′′ ( y ψ(x) )(yψ′(x) ψ(x)2 )2 ln 1 x + χ′ ( y ψ(x) )(2yψ′(x)2 ψ(x)3 − yψ′′(x) ψ(x)2 ) ln 1 x + 2 x χ′ ( y ψ(x) )(yψ′(x) ψ(x)2 ) + 1 x2 χ ( y ψ(x) ) . To make further estimates, we will write a ≈ b for any two given functions a and b to imply that the two inequalities C1a ≤ b ≤ C2a hold for all elements of the domain of a and b and for some constants C1, C2 > 0. Define f(x, y) := Lu = uxx + g(x)2uyy, using the properties of g, ψ, and χ, we then have |f(x, y)| ≈ 1 x2 + ln 1 x ( frac|ψ′′(x)|ψ(x) ) + ln 1 x (ψ′(x) ψ(x) )2 + 1 x (ψ′(x) ψ(x) ) , (5.3) and f is supported in |y| ≤ 2ψ(x). 16 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 Finite vanishing. Fix m ≥ 1 and let ψ(x) = 1 m+ 1 xm+1. Differentiating gives g(x) = ψ′(x) = xm, ψ′′(x) = mxm−1, which combined with (5.3) implies |f(x, y)| ≈ 1 x2 ln 1 x . Recall that f is supported where |y| ≤ 2ψ(x), so we can estimate the Lq norm in the ball B∫ B |f(x, y)|q dx dy . ∫ ρ 0 1 x2q ( ln 1 x )q ψ(x)dx ≈ ∫ ρ 0 1 x2q−m−1 ( ln 1 x )q dx. The right hand side is finite if and only if q < m+2 2 . We now verify that the function u belongs to the Sobolev space W 1,2 A (B), i.e. u ∈ L2(B) and∫ B |∇Au|2 dx dy = ∫ B ( |ux|2 + g(x)2|uy|2 ) dx dy <∞. Using the expressions for ux and uy and the estimates for ψ′ and ψ′′ we have for |y| ≤ 2ψ(x), |ux|2 + g(x)2|uy|2 ≈ 1 x2 ( ln 1 x )2 + g(x)2 ψ(x)2 ( ln 1 x )2 ≈ 1 x2 ( ln 1 x )2 , where for the last equality we used g = ψ′. Altogether we obtain∫ B |∇Au|2 dx dy ≈ ∫ ρ 0 1 x2 ( ln 1 x )2 ψ(x)dx ≈ ∫ ρ 0 1 x1−m ( ln 1 x )2 dx, which is finite for all m > 0. It is easy to see that u ∈ L2(B), so that u ∈W 1,2 A (B). Moreover, we have u(x, ψ(x)) = ln(1/x), so u is unbounded at the origin. Thus, for any q < m+2 2 we obtain that u is an unbounded weak solution to Lu = f with f ∈ Lq(B). On the other hand [11, Proposition 74] implies the Sobolev inequality( 1 |B| ∫ B |w|2σ ) 1 2σ ≤ Cr ( 1 |B| ∫ B |∇Aw|2 )1/2 + C ( 1 |B| ∫ B |w|2 )1/2 (5.4) for all w ∈W 1,2 0 (B), where σ′ = (m+ 2)/2. [11, Theorem 8] then implies that if u is a weak solution to Lu = f and f ∈ Lq(B) with q > m+2 2 , it is locally bounded. Infinite vanishing. We now consider the case when the function g, and therefore ψ, vanishes to infinite order at the origin. Namely, fix α > 0 and define ψ(x) := xα+1e− 1 xα , so that g(x) = ψ′(x) = αe− 1 xα + (α+ 1)xαe− 1 xα ≈ e− 1 xσ , and (ψ′(x) ψ(x) )2 ≈ ψ′′(x) ψ(x) ≈ 1 x2α+2 . EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 17 Combining this with (5.3) gives |f(x, y)| ≈ 1 x2α+2 ln 1 x , and f is supported in |y| ≤ 2ψ(x). Note that f does not belong to L∞(B) since it is unbounded at the origin, so we look for an appropriate Orlicz space for the function f . Recall the family of Young functions ΦN given in Definition 2.9. The following Orlicz-Sobolev inequality has been shown in [7] ‖w‖LΦ(B) ≤ C‖∇Aw‖L1(B) for w ∈ ( W 1,1 A ) 0 (B), if Φ = ΦN , N ≥ 1, and αN < 1. Here, B is a sufficiently small subunit metric ball centered at the origin. Letting w = v2 and using Cauchy-Schwartz inequality and Lemma 3.6 we then obtain ‖v‖LΨ(B) ≤ C‖∇Av‖L2(B) + C‖v‖L2(B) for v ∈ ( W 1,2 A ) 0 (B), (5.5) with Ψ defined by Ψ(t) = Φ(t2), i.e. Ψ(t) ≈ t2(ln t)N for all t > 1, provided Nα < 1. To make analogy to the finite type case note that (5.4) is (5.5) with Ψ(t) = t2σ, or Φ(t) = tσ. Thus, just as in the finite type case (or elliptic case), we expect all weak solutions to Lu = f to be bounded provided f belongs to a slightly smaller space than the dual of LΦ, which is LΦ̃. On the other hand, if f is in a slightly bigger space than LΦ̃ we expect there to exist an unbounded weak solution to Lu = f . We have already shown in Theorem 3.7 that every weak solution u ∈ ( W 1,2 A ) 0 (B) to Lu = f with f ∈ L∞(B) ( LΦ̃(B) is bounded. On the other hand, let u be defined by (5.2), and one can verify that u ∈W 1,2 A (B). Recall that with f = Lu we have (5.3), i.e. |f(x, y)| ≈ 1 x2α+2 ln 1 x supported in |y| ≤ 2ψ(x). We now would like to find a Young function θ so that f ∈ Lθ(B) and we expect Lθ(B) to be larger than LΦ̃(B) (analogous to q < σ′). Let θ = Φ̃M , M ≥ 1, using the estimates from [7] we have θ(s) ≤Ms1− 1 M es 1 M for s ≥ (2M)M . Therefore,∫ B θ(f(x, y)) dx dy ≈ ∫ ρ 0 θ ( 1 x2α+2 ln 1 x ) ψ(x)dx ≈ ∫ ρ 0 ( ln 1 x )1− 1 M 1 x(2α+2)(1−1/M) exp {( ln 1 x ) 1 M 1 x(2α+2)/M − 1 xα } dx. In order for this integral to be finite we must require 2α+ 2 M < α, which implies M > 2 + 2 α > 2 + 2N, 18 U. HAFEEZ, T. LAVIER, L. WILLIAMS, L. KOROBENKO EJDE-2021/82 since αN < 1. Thus f ∈ LΦ̃M (B) for M > 2 + 2N , and since M > N (i.e. LΦM ( LΦN ) we have LΦ̃N ( LΦ̃M as expected. Therefore, there exists an unbounded weak solution to Lu = f with f ∈ LΦ̃M (B). We showed that given weak solutions, u ∈ ( W 1,2 A ) 0 , to Eq. 1.1 with supB |u| ≤ ‖f‖Lψ(B), then the following Orlicz-Sobolev inequality holds: ‖u‖Lφ(B) ≤ C‖∇Au‖L2(B). We furthermore showed that given this inequality on all u ∈ (W 1,2 A )0, we are guaranteed the existence of unique weak solutions to Eq. 1.1. However, as demonstrated in Section 5, this same inequality does not ensure that supB |u| ≤ ‖f‖Lψ(B). Instead, we proved in Section 3.2, the (φ, 2) Orlicz-Sobolev inequality implies the global bound supB |u| ≤ ‖f‖L∞(B). Thus, our necessary and sufficient conditions miss each other. Counterexamples presented in Section 5 demonstrate that this miss in not merely a limitation of our proof techniques but rather, a true phenomenon within the field of degenerate elliptic partial differential equations. Acknowledgements. The first three authors would like to first and foremost thank Luda Korobenko. Without her incredible kindness, patience, and generos- ity this research would never have taken place, and our summers would have been much more dull. References [1] E. 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Shen; Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations, arXiv:1506.09203, (2016). [7] L. Korobenko, C. Rios, E. Sawyer, R. Shen; Local boundedness, maximum principles, and continuity of solutions to infinitely degenerate elliptic equations, Memoirs of the AMS, 269 (2021), No. 1311. [8] C. Léonard; Orlicz Spaces, (2007). http://leonard.perso.math.cnrs.fr/papers/Leonard-Orlicz%20spaces.pdf [9] J. Moser; On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. [10] J. Nash; Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. [11] E. Sawyer, R. L. Wheeden; Hölder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients, Memoirs of the AMS, 847 (2006). [12] W. Ungemach; Sobolev Spaces with Application to Second-Order Elliptic PDE, (2011). http://www.math.uchicago.edu/∼may/VIGRE/VIGRE2011/REUPapers/Ungemach.pdf Usman Hafeez Department of Mathematics, Reed College, Portland, OR 97202, USA Email address: usman-20@live.com EJDE-2021/82 ORLICZ-SOBOLEV INEQUALITIES AND THE DIRICHLET PROBLEM 19 Théo Lavier University of Edinburgh, Edinburgh, UK Email address: T.P.Lavier@sms.ed.ac.uk Lucas Williams Binghamton University, Binghamton, NY, USA Email address: williaml@math.binghamton.edu Lyudmila Korobenko Reed College, Portland, OR, USA Email address: korobenko@reed.edu 1. Introduction 2. Preliminaries 2.1. Subunit metric spaces 2.2. Orlicz spaces 3. Sufficiency 3.1. Existence of a unique weak solution 3.2. Global Boundedness of Weak Solutions 4. Almost necessity 5. Sharpness 5.1. Laplacian counterexample 5.2. Degenerate counterexamples Acknowledgements References