Two-scale convergence for nonlinear Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (2) ◽  
pp. 249-276 ◽  
Author(s):  
J. Casado-Díaz
Keyword(s):  
Dirichlet Problem ◽  
Periodic Structure ◽  
Limit Problem ◽  
Dirichlet Problems ◽  
Perforated Domains

The aim of the present paper is to adapt the method of two-scale convergence to the homogenization of a pseudomonotone Dirichlet problem in perforated domains with periodic structure. The limit problem and a corrector result are obtained.

scholarly journals Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

1995 ◽  
Vol 125 (1) ◽  
pp. 99-114 ◽  
Author(s):  
Gianni Dal Maso ◽  
Annalisa Malusa
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Radon Measure ◽  
Boundary Value ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Perforated Domains ◽  
Positive Radon Measure

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

scholarly journals On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

2021 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Luigi Provenzano
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Lipschitz Domain ◽  
Spectral Properties ◽  
Explicit Representation ◽  
Lipschitz Domains ◽  
Dirichlet Problems ◽  
Trace Space ◽  
In Series ◽  
Definition Of

AbstractWe consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains

2018 ◽  
Vol 149 (2) ◽  
pp. 533-560
Author(s):  
Patricio Felmer ◽  
Erwin Topp
Keyword(s):  
Dirichlet Problem ◽  
Approximation Scheme ◽  
Linear Operators ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Uniqueness Of Solutions ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Continuous Boundary ◽  
Extra Assumption

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

scholarly journals The asymptotic Dirichlet problems on manifolds with unbounded negative curvature

2018 ◽  
Vol 167 (01) ◽  
pp. 133-157
Author(s):  
RAN JI
Keyword(s):  
Dirichlet Problem ◽  
Negative Curvature ◽  
Probabilistic Method ◽  
Martin Boundary ◽  
Dirichlet Problems ◽  
Curvature Condition ◽  
New Approach ◽  
Harnack Inequalities ◽  
Geometric Boundary ◽  
Analytical Proof

AbstractElton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature condition −Ce(2−η)r(x) ≤ KM(x) ≤ −1 with η > 0. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition −Ce(2/3−η)r(x) ≤ KM(x) ≤ −1 with η > 0. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of M. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proof is a modification of an argument due to M. T. Anderson and R. Schoen.

scholarly journals A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

2019 ◽  
Vol 53 (3) ◽  
pp. 987-1003 ◽  
Author(s):  
Claudio Canuto ◽  
Ricardo H. Nochetto ◽  
Rob P. Stevenson ◽  
Marco Verani
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
Galerkin Approximation ◽  
Error Reduction ◽  
Two Dimensional ◽  
Dirichlet Problems ◽  
Polynomial Degree ◽  
Unit Square ◽  
The Poisson Equation ◽  
Dimensional Unit

Both practice and analysis of p-FEMs and adaptive hp-FEMs raise the question what increment in the current polynomial degree p guarantees a p-independent reduction of the Galerkin error. We answer this question for the p-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree p. We show that an increment proportional to p yields a p-robust error reduction and provide computational evidence that a constant increment does not.

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS

1994 ◽  
Vol 04 (03) ◽  
pp. 373-407 ◽  
Author(s):  
GIANNI DAL MASO ◽  
ADRIANA GARRONI
Keyword(s):  
Adjoint Operator ◽  
Arbitrary Sequence ◽  
Symmetric Part ◽  
Dirichlet Problems ◽  
Limit Measure ◽  
Compactness Result ◽  
Perforated Domains ◽  
Measurable Coefficients ◽  
Open Set ◽  
Γ Convergence

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions [Formula: see text] of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in [Formula: see text] to the unique solution [Formula: see text] of the problem [Formula: see text] When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions [Formula: see text] of the equations [Formula: see text] where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

scholarly journals About the Dirichlet Boundary Value Problem using Clifford Algebras

2018 ◽  
Vol 15 ◽  
pp. 8098-8119
Author(s):  
Johan Ceballos
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Clifford Algebras ◽  
Dirichlet Boundary ◽  
Relevant Literature ◽  
Monogenic Functions ◽  
Dirichlet Boundary Value Problem ◽  
Dirichlet Problems ◽  
Initial Value ◽  
Do So

This paper reviews and summarizes the relevant literature on Dirichlet problems for monogenic functions on classic Clifford Algebras and the Clifford algebras depending on parameters on. Furthermore, our aim is to explore the properties when extending the problem to and, illustrating it using the concept of fibres. To do so, we explore ways in which the Dirichlet problem can be written in matrix form, using the elements of a Clifford's base. We introduce an algorithm for finding explicit expressions for monogenic functions for Dirichlet problems using matrices in Finally, we illustrate how to solve an initial value problem related to a fibre.

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