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Sobolev embeddings into Orlicz spaces on domains in the Eu-clidean space or, more generally, on Riemannian manifolds are considered. Highly irregular domains where the optimal degree of integrability of a func-tion may be lower than the one of its gradient are focused. A necessary and sufficient condition for the validity of the relevant embeddings is established in terms of the isocapacitary function of the domain. Compact embeddings are discussed as well. Sufficient conditions involving the isoperimetric function of the domain are derived as a by-product., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) Prin 2017 "Direct and inverse problems for partial differential equations: theoretical aspects and applications"; GNAMPA of the Italian INdAM -National Institute of High Mathematics [201758MTR2]

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in Rn. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace. (C) 2019 Elsevier Ltd. All rights reserved., Funding Agencies|Research Project of the Italian Ministry of University and Research (MIUR) Prin 2015 "Partial differential equations and related analytic-geometric inequalities" [2015HY8JCC]; GNAMPA of the Italian INdAM -National Institute of High Mathematics; RUDN University Program 5-100, Russia

Second-order estimates are established for solutions to the p-Laplace system with right-hand side in L-2. The nonlinear expression of the gradient under the divergence operator is shown to belong to W-1,W-2, and hence to enjoy the best possible degree of regularity. Moreover, its norm in 1471,2 is proved to be equivalent to the norm of the right-hand side in L-2. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well. (C) 2019 Elsevier Masson SAS. All rights reserved., Funding Agencies|Italian Ministry of University and Research (MIUR)Ministero dell Istruzione, dell Universita e della Ricerca (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM -National Institute of High Mathematics; RUDN University Program [5-100]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]