1,720,988 research outputs found
The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity
Full text linkWe investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions
Principal eigenvalues for k-Hessian operators by maximum principle methods
Full text linkFor fully nonlinear k-Hessian operators on bounded strictly (k - 1)-convex domains Ω of RN, a characterization of the principal eigenvalue associated to a k-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone Σk ⊂ S(N) which is an elliptic set in the sense of Krylov [23] which corresponds to using k-convex functions as admissibility constraints in the formulation of viscosity subsolutions and supersolutions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property which results from the establishment of a global Hölder estimate for the unique k-convex solutions of the approximating equations
Existence of the Principal Eigenvalue for Cooperative Elliptic Systems in a General Domain
No full textExistence through convexity for the truncated Laplacians
Full text linkWe study the Dirichlet problem on a bounded convex domain of RN, with zero boundary data, for truncated Laplacians Pk±, which are degenerate elliptic operators, for k< N, defined by the upper and respectively lower partial sum of k eigenvalues of the Hessian matrix. We establish a necessary and sufficient condition (Theorem 1) in terms of the “flatness” of domains for existence of a solution for general inhomogeneous term. This result, in particular, shows that the strict convexity of the domain is sufficient for the solvability of the Dirichlet problem. The result and related ideas are applied to the solvability of the Dirichlet problem for the operator Pk+ with lower order term when the domain is strictly convex and the existence of principal eigenfunctions for the operator P1+. An existence theorem is presented with regard to the principal eigenvalue for the Dirichlet problem with zero-th order term for the operator P1+. A nonexistence result is established for the operator Pk+ with first order term when the domain has a boundary portion which is nearly flat. Furthermore, when the domain is a ball, we study the Dirichlet problem, with a constant inhomogeneous term and a possibly sign-changing first order term, and the associated eigenvalue problem
Fractional truncated Laplacians: representation formula, fundamental solutions and applications
Full text linkWe introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called “fundamental solutions”. This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to “understand” what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differ significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian
Existence Issues for a Large Class of Degenerate Elliptic Equations with Nonlinear Hamiltonians
No full textFor degenerate elliptic equations with a nonlinear gradient term H, in bounded uniformly convex domains Ω, we give sufficient conditions for the existence and uniqueness of solutions in terms of the size of Ω, of the forcing term f and of H. The results apply to a wide class of equations, having as principal part significant examples, e.g. linear degenerate operators, weighted partial trace operators and the homogeneous Monge-Ampère operator
Mixed boundary value problems for fully nonlinear degenerate or singular equations
No full textWe prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Holder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Holder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems. (c) 2022 Elsevier Ltd. All rights reserved
Towards a reversed Faber–Krahn inequality for the truncated Laplacian
Full text linkWe consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator P1+ mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian
C1,γ regularity for singular or degenerate fully nonlinear equations and applications
Full text linkIn this note, we prove C1,γ regularity for solutions of some fully nonlinear degenerate elliptic equations with “superlinear” and “subquadratic” Hamiltonian terms. As an application, we complete the results of Birindelli et al. (ESAIM Control Optim Calc Var, 2019. https://doi.org/10.1051/cocv/2018070) concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function
Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a liouville type theorem
Full text linkWe prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to constants, of the ergodic functions. The results are obtained by means of a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators
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