scholarly journals Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

2019 ◽  
Vol 150 (1) ◽  
pp. 475-495 ◽  
Author(s):  
Begoña Barrios ◽  
Maria Medina
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Classical Case ◽  
Parabolic Problems ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Local Version ◽  
Comparison Results ◽  
Non Local

AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

scholarly journals Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

2019 ◽  
Vol 39 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì ◽  
Angela Sciammetta
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Mixed Boundary Conditions ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Differential Problem ◽  
Nonlinear Elliptic

In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.

Non-linear mixed boundary value problems for parabolic partial differential equations

1985 ◽  
Vol 100 (3-4) ◽  
pp. 219-235 ◽  
Author(s):  
Joelle Bailet-Intissar
Keyword(s):  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Open Subset ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Parabolic Problems ◽  
Parabolic Partial Differential Equations ◽  
Sufficient Condition ◽  
Non Linear ◽  
Bounded Open Subset

SynopsisA sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.

scholarly journals Eigenvalues of the Laplacian with Neumann boundary conditions

1985 ◽  
Vol 26 (3) ◽  
pp. 293-309 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Summation Formula ◽  
Laplacian Operator ◽  
Eigenvalues Of The Laplacian ◽  
Annular Sector ◽  
Simply Connected

AbstractVarious grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

scholarly journals Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

2020 ◽  
Vol 20 (3) ◽  
pp. 633-650
Author(s):  
Wolfgang Arendt ◽  
A. F. M. ter Elst ◽  
Jochen Glück
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Lipschitz Continuous ◽  
Various Boundary Conditions ◽  
Measurable Coefficients ◽  
Strict Positivity ◽  
Robin Boundary

AbstractWe consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies {v(x)\geq\delta>0} for all {x\in\overline{\Omega}}, even if the boundary {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map {L_{p}(\Omega)} into {C(\overline{\Omega})}. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

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