An interior estimate of the gradient of a solution to the Dirichlet problem for equations of curvature type

1995 ◽  
Vol 77 (4) ◽  
pp. 3391-3397
Author(s):  
G. V. Yakunina
Keyword(s):  
Dirichlet Problem ◽  
Interior Estimate ◽  
Curvature Type

scholarly journals On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

2020 ◽  
Vol 59 (4) ◽  
Author(s):  
Leonardo Bonorino ◽  
Jean-Baptiste Casteras ◽  
Patricia Klaser ◽  
Jaime Ripoll ◽  
Miriam Telichevesky
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mean Curvature ◽  
Partial Differential ◽  
Curvature Type

scholarly journals Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

2019 ◽  
Vol 58 (6) ◽  
Author(s):  
Yunelsy N. Alvarez ◽  
Ricardo Sa Earp
Keyword(s):  
Dirichlet Problem ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Necessary Condition ◽  
Dirichlet Problems ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Sectional Curvatures ◽  
Curvature Type

Abstract It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $${{\,\mathrm{\mathbb {R}}\,}}^n$$Rn with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $$ M^n \times {{\,\mathrm{\mathbb {R}}\,}}$$Mn×R. Precisely, given a $$\mathscr {C}^2$$C2 bounded domain $$\Omega $$Ω in M and a function $$ H = H (x, z) $$H=H(x,z) continuous in $$\overline{\Omega }\times {{\,\mathrm{\mathbb {R}}\,}}$$Ω¯×R and non-decreasing in the variable z, we prove that the strong Serrin condition$$(n-1)\mathcal {H}_{\partial \Omega }(y)\ge n\sup \limits _{z\in {{\,\mathrm{\mathbb {R}}\,}}}\left| H(y,z) \right| \ \forall \ y\in \partial \Omega $$(n-1)H∂Ω(y)≥nsupz∈RH(y,z)∀y∈∂Ω, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins–Serrin and Serrin type sharp solvability criteria.

Solvability of one nonlocal Dirichlet problem for the Poisson equation

2019 ◽  
Vol 50 (1) ◽  
pp. 67-88
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
The Poisson Equation

scholarly journals Existence of strong solutions to the Dirichlet problem for the Griffith energy

2019 ◽  
Vol 58 (4) ◽  
Author(s):  
Antonin Chambolle ◽  
Vito Crismale
Keyword(s):  
Dirichlet Problem ◽  
Strong Solutions

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

2021 ◽  
Vol 279 ◽  
pp. 1-9
Author(s):  
Inka Schnieders ◽  
Guido Sweers
Keyword(s):  
Dirichlet Problem

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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