Homogeneous Dirichlet Boundary Conditions Research Articles

Normalized concentrating solutions to nonlinear elliptic problems

We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem{−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω is either the whole space RN or a bounded smooth domain of RN, in which case we assume V≡0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, 1<p<N+2N−2 if N≥3 and p>1 if N=1,2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Ω as the prescribed mass ρ is either small (when p<1+4N) or large (when p>1+4N) or it approaches some critical threshold (when p=1+4N).

Well-posedness of renormalized solutions for a stochastic p -Laplace equation with L^1 -initial data

We consider a $ p $-Laplace evolution problem with stochastic forcing on a bounded domain $ D\subset\mathbb{R}^d $ with homogeneous Dirichlet boundary conditions for $ 1&lt;p&lt;\infty $. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data $ u_0 $ under consideration. Due to the poor regularity of the initial data, estimates in $ W^{1,p}_0(D) $ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

Positive solutions of superlinear elliptic problems with discontinuous non-linearities

We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition, a parameter and a discontinuous non-linearity. The positive parameter appears as a multiplicative term in the non-linearity, and the problem has a zero solution for any value of the parameter. The non-linearity has superlinear growth at infinity. We prove the existence of positive solutions by a topological method.

Uniqueness and regularity for the 3D Boussinesq system with damping

This paper is concerned with the Boussinesq system with a damping term and the homogeneous Dirichlet boundary conditions in 3D bounded domains. For a certain range of parameters, we prove that the weak solution is unique if the temperature belongs to $$L^{\infty }(0,T;L^{3}(\Omega ))$$ . Also, the global existence of strong solutions to the problem is proved.

Pseudospectral Time-Domain (PSTD) Methods for the Wave Equation: Realizing Boundary Conditions with Discrete Sine and Cosine Transforms

Pseudospectral time domain (PSTD) methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology. The use of the Fourier collocation spectral method in particular has many computational advantages. However, the use of a discrete Fourier basis is also inherently restricted to solving problems with periodic boundary conditions. Here, a family of spectral collocation methods based on the use of a sine or cosine basis is described. These retain the computational advantages of the Fourier collocation method but instead allow homogeneous Dirichlet (sound-soft) and Neumann (sound-hard) boundary conditions to be imposed. The basis function weights are computed numerically using the discrete sine and cosine transforms, which can be implemented using [Formula: see text] operations analogous to the fast Fourier transform. Practical details of how to implement spectral methods using discrete sine and cosine transforms are provided. The technique is then illustrated through the solution of the wave equation in a rectangular domain subject to different combinations of boundary conditions. The extension to boundaries with arbitrary real reflection coefficients or boundaries that are nonreflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.

Fluctuations Around a Homogenised Semilinear Random PDE

We consider a semilinear parabolic partial differential equation in mathbf{R}_+times [0,1]^d, where d=1, 2 or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension d=1, that rescaled difference converges as one might expect to a centred Ornstein–Uhlenbeck process. However, in dimension d=2, the limit is a non-centred Gaussian process, while in dimension d=3, before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.

Periodic travelling wave solutions for a reaction-diffusion system on landscape fitted domains

In this article, we propose a new landscape fitted domain construction and its boundary treatment of periodic travelling wave solutions for a diffusive predator-prey system with landscape features. The proposed method uses the distance function based on an obstacle. The landscape fitted domain is defined as a region whose distance from the obstacle is positive and less than a pre-defined distance. At the exterior boundary of the domain, we use the zero-Neumann boundary condition and define the boundary value from the bilinearly interpolated value in the normal direction of the distance function. At the interior boundary, we use the homogeneous Dirichlet boundary condition. Typically, reaction-diffusion systems are numerically solved on rectangular domains. However, in the case of periodic travelling wave solutions, the boundary treatment is critical because it may result in unexpected chaotic pattern. To avoid this unwanted chaotic behavior, we need to use sufficiently large computational domain to minimize the boundary treatment effect. Using the proposed method, we can get accurate results even though we use relatively small domain sizes.

Diffusive Epidemiological Predator–Prey Models with Ratio-Dependent Functional Responses and Nonlinear Incidence Rate

In this paper, when the predator breeds only by the prey, the relationship among the two classes of prey (susceptible and infected prey) and the predator is represented by using an epidemic model with nonlinear incidence and the response function depending on the density of the three individuals under homogeneous Dirichlet boundary conditions describing a hostile environment at its boundary. For this model, the coexistence steady state of three interacting species is mainly investigated, and the global stabilities of the extinction of all species and the survival of only healthy prey are discussed. Additionally, the conditions of the non-coexistence of three species are established. Finally, an ecological interpretation is also presented based on the results.

On some classes of generalized Schrödinger equations

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + ∑ i = 2 m $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim V λ i pairs of nontrivial solutions if a parameter involved in the equation is large enough, where V λ i denotes the eigenspace associated to the i-th eigenvalue λ i of laplacian operator with homogeneous Dirichlet boundary condition.

Existence and Multiplicity of Solutions for the Equation with Nonlocal Integrodifferential Operator

We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator $${\mathcal {L}}_K$$ $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$\Omega $$ be an open-bounded set of $${\mathbb {R}}^n$$ with continuous boundary, $$\lambda \in {\mathbb {R}}$$ is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with $$\lambda <\lambda _1$$ , where $$\lambda _1$$ is the first eigenvalue of the nonlocal operator $$-{\mathcal {L}}_K$$ with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with $$\lambda \ge \lambda _1$$ . As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$0<\alpha <1$$ . The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.

The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains

The Stokes resolvent problem lambda u - Delta u + nabla phi = f with {text {div}}(u) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of mathrm {L}^2_{sigma } (Omega ) ni f mapsto phi in mathrm {L}^2 (Omega ) decays like |lambda |^{- 1 / 2} which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like |lambda |^{- alpha } for 0 le alpha le 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain Omega is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side f in mathrm {L}^2 (Omega ; {mathbb {C}}^d) admit mathrm {H}^2-regularity and further prove localized mathrm {H}^2-estimates for the Stokes resolvent problem. By a generalized version of Shen’s mathrm {L}^p-extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in mathrm {L}^p (Omega ; {mathbb {C}}^d) for 2d / (d + 2)< p < 2d / (d - 2) (with 1< p < infty if d = 2). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.

Compactness and dichotomy in nonlocal shape optimization

AbstractWe prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration‐compactness principle, we prove that either an optimal shape exists or there exists a minimizing sequence consisting of two “pieces” whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.

Optimization of the shape of regions supporting boundary conditions

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution $$u_\varOmega $$ to a Laplace equation in a domain $$\varOmega $$ ; the boundary $$\partial \varOmega $$ is divided into three parts $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative $$J^\prime (\varOmega )(\theta )$$ of a general objective function $$J(\varOmega )$$ of the domain is calculated in the framework of Hadamard’s method when the considered deformations $$\theta $$ are allowed to modify the geometry of $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ (i.e. $$\theta $$ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of $$u_\varOmega $$ near the boundaries of the regions $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ . For this reason, in particular, $$J^\prime (\varOmega )$$ is difficult to calculate and to evaluate numerically when the transition $$\overline{\varGamma _D} \cap {\overline{\varGamma }}$$ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary $$\partial \varOmega $$ with coefficients depending on a small parameter $$\varepsilon $$ . We prove the consistency of this approach in our model context: the approximate objective function $$J_\varepsilon (\varOmega )$$ and its shape derivative converge to their exact counterparts as $$\varepsilon $$ vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.

A class of nonlinear parabolic equations with anisotropic nonstandard growth conditions

In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with the solution. Second, we give two new blow-up criteria that one of them is obtained to remove the exponent restriction by using an extended form of the concavity method and the other one is a blow-up result of weak solutions with arbitrarily high initial energy. Moreover, an extinction result is also showed by establishing a new auxiliary lemma. The present results of this paper complement and improve a recent paper investigated by Liu et al. [J. Math. Phys. 59, 121504 (2018)].

Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

In this paper we obtain comparison results for the quasilinear equation −Δp,xu−uyy=f with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable x, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in y, and the proof of a comparison principle for the discrete version of the auxiliary problem AU−Uyy≤∫0sf, where AU=(nωn1/ns1/n′)p(−Uss)p−1. We show that this operator is T-accretive in L∞. We extend our results for −Δp,x to general operators of the form −div(a(|∇xu|)∇xu) where a is non-decreasing and behaves like |⋅|p−2 at infinity.

Existence of Solution for Nonlocal Heterogeneous Elliptic Problems

We consider a class of a nonlocal heterogeneous elliptic problem of type $$\begin{aligned} -M(|u|_{q}^{q})\,div(a(x)\nabla u)=g(x,u) \end{aligned}$$ with a homogeneous Dirichlet boundary condition. Under different assumptions on the function g, we establish two existence theorems for this problem by using, respectively, the Schauder and Tychonoff fixed point theorems. Also, we give an example for each theorem.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces

We introduce the transportation-annihilation distanceW_p^sharp between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space. We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow. For the Dirichlet heat flow, moreover, we establish a gradient flow interpretation within a suitable space of charged probabilities. In order to prove this, we will work with the doubling of the open set, the space obtained by gluing together two copies of it along the boundary.

Finite element approximation of elliptic homogenization problems in nondivergence-form

We use uniform W2,p estimates to obtain corrector results for periodic homogenization problems of the form A(x/ε):D2uε = f subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.