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 Dirichlet problem for Poisson equations in Jordan domains

2018 ◽  
Vol 32 ◽  
pp. 30-41
Author(s):  
Vladimir Gutlyanskii ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Continuous Functions ◽  
Logarithmic Capacity ◽  
Hölder Continuous ◽  
Arbitrary Boundary ◽  
Poisson Equations ◽  
Local Properties ◽  
Continuous Boundary ◽  
Angular Limits

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

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.

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 Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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.

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