scholarly journals Comparison results for solutions to p-Laplace equations with Robin boundary conditions

2021 ◽  
Author(s):  
Vincenzo Amato ◽  
Andrea Gentile ◽  
Alba Lia Masiello
Keyword(s):  
Boundary Conditions ◽  
Laplace Operator ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Robin Boundary Conditions ◽  
Planar Case ◽  
Laplace Equations ◽  
Comparison Results ◽  
Robin Boundary ◽  
Appl Math

AbstractIn the last decades, comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in Alvino et al. (Commun Pure Appl Math, to appear) to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in Alvino et al. (to appear) only in the planar case, holds true in any dimension if p is sufficiently small.

scholarly journals The asymptotically linear Hénon problem

2020 ◽  
pp. 2050042
Author(s):  
Anna Lisa Amadori
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Radial Solutions ◽  
Asymptotically Linear ◽  
Planar Case ◽  
Asymptotic Profile ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent [Formula: see text] is close to [Formula: see text]. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle [Formula: see text]. By considerations based on the Morse index we see that, depending on the values of [Formula: see text] and [Formula: see text], such least energy solutions can be radial, or nonradial and different one from another.

scholarly journals On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional Orders

2020 ◽  
Vol 43 (6) ◽  
pp. 4089-4106
Author(s):  
Rafał Kamocki
Keyword(s):  
Boundary Conditions ◽  
Differential Inclusion ◽  
Laplace Operator ◽  
Spectral Decomposition ◽  
Convex Analysis ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Fenchel Transform ◽  
The Laplace Operator ◽  
Fractional Orders

Abstract In this paper, we investigate a nonlinear differential inclusion with Dirichlet boundary conditions containing a weak Laplace operator of fractional orders (defined via the spectral decomposition of the Laplace operator $$-{\varDelta }$$ - Δ under Dirichlet boundary conditions). Using variational methods, we characterize solutions of such a problem. Our approach is based on tools from convex analysis (properties of a Legendre–Fenchel transform).

ASYMPTOTIC FORMULAS FOR DISCRETE EIGENVALUE PROBLEMS IN LIOUVILLE NORMAL FORM

2001 ◽  
Vol 11 (01) ◽  
pp. 43-56 ◽  
Author(s):  
FRANK R. de HOOG ◽  
ROBERT S. ANDERSSEN
Keyword(s):  
Boundary Conditions ◽  
Normal Forms ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
General Boundary ◽  
Asymptotic Formulas ◽  
General Boundary Conditions ◽  
Discrete Eigenvalue ◽  
Robin Boundary

In the analysis of both continuous and discrete eigenvalue problems, asymptotic formulas play a central and crucial role. For example, they have been fundamental in the derivation of results about the inversion of the free oscillation problem of the Earth and related inverse eigenvalue problems, the computation of uniformly valid eigenvalues approximations, the proof of results about the behavior of the eigenvalues of Sturm–Liouville problems with discontinuous coefficients, and the construction of a counterexample to the Backus–Gilbert conjecture. Useful formulas are available for continuous eigenvalue problems with general boundary conditions as well as for discrete eigenvalue problems with Dirichlet boundary condition. The purpose of this paper is the construction of asymptotic formulas for discrete eigenvalue problems with general boundary conditions. The motivation is the computation of uniformly valid eigenvalue approximations. It is now widely accepted that the algebraic correction procedure, first proposed by Paine et al.,13 is one of the simplest methods for computing uniformly valid approximations to a sequence of eigenvalues of a continuous eigenvalue problem in Liouville normal form.8 This relates to the fact that, for Liouville normal forms with Dirichlet boundary conditions, it is not too difficult to prove that such procedures yield, under quite weak regularity conditions, uniformly valid O(h2) approximations. For Liouville normal forms with general boundary conditions, the corresponding error analysis is technically more challenging. Now it is necessary to have, for such Liouville normal forms, higher order accurate asymptotic formulas for the eigenvalues and eigenfunctions of their continuous and discrete counterparts. Assuming that such asymptotic formulas are available, it has been shown1 how uniformly valid O(h2) results could be established for the application of the algebraic correction procedure to Liouville normal forms with general boundary conditions. Algorithmically, this methodology represents an efficient procedure for determining uniformly valid approximations to sequences of eigenvalues, even though it is more complex than for Liouville normal forms with Dirichlet boundary conditions. As well as giving a brief review of the subject for general (Robin) boundary conditions, this paper sketches proofs for the asymptotic formulas, for Robin boundary conditions, which are required in order to construct the mentioned O(h2) results.

Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along (n − 1)-dimensional manifolds

2014 ◽  
Vol 89 (1) ◽  
pp. 11-15 ◽  
Author(s):  
D. Gómez ◽  
M. Lobo ◽  
E. Pérez ◽  
A. V. Podolskiy ◽  
T. A. Shaposhnikova
Keyword(s):  
Boundary Conditions ◽  
Laplace Operator ◽  
Robin Boundary Conditions ◽  
Robin Boundary
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