Homogeneous Dirichlet Problem Research Articles

The long wavelength limit of periodic solutions of water wave models

AbstractThe present essay is concerned with providing rigorous justification of a long‐standing practice in numerical simulation of partial differential equations. Theory often sets initial‐value problems on all of or . If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach.

Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

Regularity results for an anisotropic nonlinear Dirichlet problem

In this paper we consider the homogeneous Dirichlet problem associated to the model equation − div ⁡ ( a ( x ) | ∇u | p − 2 ∇u ) − div ⁡ ( | u | ( r − 1 ) q + 1 | ∇u | q − 2 ∇u ) = f in Ω , where Ω is a bounded open subset of 2, $ ]]> R N , N > 2 , 1<q<p<N, \\frac {1}{q'} $ ]]> r > 1 q ′ and f belongs to 1 $ ]]> L m ( Ω ) , m > 1 . We give some regularity results according to the values of the exponents ( r − 1 ) q + 1 and m in suitable ranges depending on N , p and q .

High-Precision Calculation Using the Method of Analytical Regularization for the Cut-Off Wave Numbers for Waveguides of Arbitrary Cross Sections with Inner Conductors

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. Concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter provide a secure basis for formulating the spectral problem, and the Method of Analytical Regularization is employed to implement an effective algorithm. The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries. A highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution is developed. The mathematical correctness of the problem statement, the method, and the ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provide a secure basis for treating these as “benchmark solutions”. In this paper, we use this new approach to validate previously obtained results against our benchmark solutions. Furthermore, we demonstrate its universality in solving some new problems, which are barely accessible by existing methods.

Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed. As an example, an explicit form of the solution of the homogeneous Dirichlet problem for the inhomogeneous polyharmonic equation with the simplest polynomial right-hand side is found.

Erratum to: Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain

Erratum to: Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain

Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain

Comparison of Approximate and Numerical Methods for Solving the Homogeneous Dirichlet Problem for the Helmholtz Operator in a Two-Dimensional Domain

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

On Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

The paper gives an explicit representation of the Green’s function of the Dirichlet boundary value problem for the polyharmonic equation in the unit ball. The solution of the homogeneous Dirichlet problem is found. An example of solving the homogeneous Dirichlet problem with the simplest polynomial right-hand side of the polyharmonic equation is given.

Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of Rn,n≥3 and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri–Coron and demands new constructions and extra ideas.

Periodic unfolding for anisotropically bounded sequences

This paper is focused on the asymptotic behavior of sequences of functions, whose partial derivatives estimates in one or more directions are highly contrasted with respect to the periodic parameter ε. In particular, a direct application for the homogenization of a homogeneous Dirichlet problem defined on an anisotropic structure is presented. In general, the obtained results can be applied to thin structures where the behavior is different according to the observed direction.

Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $$(-\Delta )^s$$ . We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $$C^s$$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

Fractional‐order operators on nonsmooth domains

AbstractThe fractional Laplacian , , and its generalizations to variable‐coefficient ‐order pseudodifferential operators , are studied in ‐Sobolev spaces of Bessel‐potential type . For a bounded open set , consider the homogeneous Dirichlet problem: in , in . We find the regularity of solutions and determine the exact Dirichlet domain (the space of solutions with ) in cases where has limited smoothness , for , . Earlier, the regularity and Dirichlet domains were determined for smooth by the second author, and the regularity was found in low‐order Hölder spaces for by Ros‐Oton and Serra. The ‐results obtained now when are new, even for . In detail, the spaces are identified as ‐transmission spaces , exhibiting estimates in terms of near the boundary.The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.

Resolvents for fractional-order operators with nonhomogeneous local boundary conditions

For 2a-order strongly elliptic operators P generalizing (−Δ)a, 0<a<1, the homogeneous Dirichlet problem on a bounded open set Ω⊂Rn has been widely studied. Pseudodifferential methods have been applied by the present author when Ω is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of Lq-Sobolev spaces Hqs for 1<q<∞, when Ω is Cτ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the Lp-Dirichlet realizations of P and P⁎, showing that there are finite-dimensional kernels and cokernels lying in daCα(Ω‾) with suitable α>0, d(x)=dist(x,∂Ω). Similar results are established for P−λI, λ∈C. The solution spaces equal a-transmission spaces Hqa(t)(Ω‾).Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da−1)|∂Ω. They are solvable in the larger spaces Hq(a−1)(t)(Ω‾). Furthermore, the nonhomogeneous problem with a spectral parameter λ∈C,Pu−λu=f in Ω,u=0 in Rn∖Ω,(u/da−1)|∂Ω=φ on ∂Ω, is for q<(1−a)−1 shown to be uniquely resp. Fredholm solvable when λ is in the resolvent set resp. the spectrum of the L2-Dirichlet realization.The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter t, both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace (u(x,t)/da−1(x))|x∈∂Ω is prescribed.

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equationut−∑i=1NDxi(|Dxiu|pi(x,t)−2Dxiu)=f(x,t) in the cylinder Ω×(0,T), where Ω⊂RN, N≥2, is a parallelepiped. The exponents of nonlinearity pi are given Lipschitz-continuous functions. It is shown that if pi(x,t)>2NN+2,μ=supQT⁡maxi⁡pi(x,t)mini⁡pi(x,t)<1+1N,|Dxiu0|max⁡{pi(⋅,0),2}∈L1(Ω),f∈L2(0,T;W01,2(Ω)), then the problem has a unique solution u∈C([0,T];L2(Ω)) with |Dxiu|pi∈L∞(0,T;L1(Ω)), ut∈L2(QT). Moreover,|Dxiu|pi+r∈L1(QT)with some r=r(μ,N)>0,|Dxiu|pi−22Dxiu∈W1,2(QT). The assertions remain true for a smooth domain Ω if pi=2 on the lateral boundary of QT.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Stability for Evolution Equations with Variable Growth

We study the homogeneous Dirichlet problem for the evolution p(x, t)-Laplacian with the nonlinear source ut-div|∇u|p(x,t)-2∇u=f(x,t,u),(x,t)∈Q=Ω×(0,T).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t-{\ ext {div}}\\left( |\ abla u|^{p(x,t)-2}\ abla u\\right) =f(x,t,u),\\quad (x,t)\\in Q=\\Omega \ imes (0,T). \\end{aligned}$$\\end{document}Here, Omega subset {mathbb {R}}^n is a bounded domain, nge 2, and p(x,!t) is a given function p(cdot ):Qmapsto (frac{2n}{n+2},p^+], p^+<infty . It is shown that the solution is stable with respect to perturbations of the exponent p(x, t), the nonlinear source f(x, t, u), and the initial datum. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the exponent p(x, t) and the data u(x, 0), f. Estimates on the rate of convergence of solutions of perturbed problems to the solution of the limit problem are derived.

Existence of Hölder Continuous Solutions for a Class of Degenerate Fourth-Order Elliptic Equations

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of a homogeneous Dirichlet problem related to a class of nonlinear fourth-order elliptic equations with strengthened degenerate ellipticity condition.

An analogue of the Lyapunov inequality for an ordinary second-order differential equation with a fractional derivative and a variable coefficient

This paper studies an ordinary second-order differential equation with a fractional differentiation operator in the sense of Riemann-Liouville with a variable coefficient. We use the Green’s function’s method to find a representation of the solution of the Dirichlet problem for the equation under consideration when the solvability condition is satisfied. Green’s function to the problem is constructed in terms of the fundamental solution of the equation under study and its properties are proved. The necessary integral condition for the existence of a nontrivial solution to the homogeneous Dirichlet problem, called an analogue of the Lyapunov inequality, is found.

Non-local tug-of-war with noise for the geometric

This paper concerns the fractional ${\mathrm{\bf p}}$-Laplace operator $\Delta_{\mathrm{\bf p}}^s$ in non-divergence form, which has been introduced in [2]. For any ${\mathrm{\bf p}}\in [2,\infty)$ and $s\in (\frac{1}{2},1)$, we first define two families of non-local, non-linear averaging operators, parametrized by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb R^N\to \mathbb R$. We prove that $\Delta_{\mathrm{\bf p}}^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb R^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_{\mathrm{\bf p}}^s$, posed in a domain ${\mathcal{D}}$ satisfying the external cone condition and subject to bounded, uniformly continuous data on $\mathbb R^N\setminus {\mathcal{D}}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on ${\mathrm{\bf p}}$ and $N$, and whose $\epsilon$-tip has been removed.