We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed domain. Then we show that the Dirichlet eigenvalues corresponding to the family of perturbed domains converge to the Dirichlet eigenvalues corresponding to the unperturbed domain. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lamé system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre-Hadamard ellipticity condition.
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.
We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}Δu + u = f(u){\text{ in }}Ω,{\text{ }}u > 0{\text{ in }}Ω,{\text{ }}u = 0{\text{ on }}\partial Ω, \] where $Ω$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial Ω\in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial Ω$.
Given a smooth Riemannian manifold (M,g)we investigate the existence of positive solutions to a singularly perturbed supercritical elliptic equation which concentrate at some submanifold of M. We obtain a posive answer for some manifolds, which include warped products.
Let $Ω$ be a bounded domain in $\mathbb{R}^{d}$ $\left( d\geq 2\right) $ pretty regular. We solve the variational Dirichlet problem for a class of quasi-linear elliptic systems.
We show that the number of low energy solutions of a double singularly perturbed Schroedinger Maxwell system type on a smooth 3 dimensional manifold (M,g) depends on the topological properties of the manifold. The result is obtained via Lusternik Schnirelmann category theory.
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
In this note we complete the study made in a previous paper on a Kirchhoff type equation with a Berestycki-Lions nonlinearity. We also correct Theorem 0.6 inside.
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number ε and the symmetric metric g. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $Ω\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.
We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $Ω\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.
We consider a sufficiently regular bounded open connected subset $Ω$ of $\mathbb{R}^n$ such that $0 \in Ω$ and such that $\mathbb{R}^n \setminus \clΩ$ is connected. Then we choose a point $w \in ]0,1[^n$. If $ε$ is a small positive real number, then we define the periodically perforated domain $T(ε) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+εΩ+z)$. For each small positive $ε$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(ε)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+εΩ$, and we denote the unique peri…
In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operat…
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(λ(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior at infinity.
This paper studies a nonlinear Dirichlet problem for the $p$-Laplacian operator with nonlinearity consisting of power components. The problem under consideration can be though of as a perturbation of the Ambrosetti-Brezis-Cerami problem with concave-convex nonlinearity. The combined effect of power components in the perturbed nonlinearity allows to establish a higher order multiplicity of positive solutions. We study properties of the perturbed energy functional and prove the existence of three positive solutions to the problem at hand.
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.
Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_ε]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a particular Dir…
Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables.
By means of a penalization scheme due to del Pino and Felmer, we prove the existence of single-peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness. We don't require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity. Compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the Pucci-Serrin variational identity plays an important role.
We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $Ω\subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincaré-Sobolev inequalities.
We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetildeΩ \subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.
In this article we are interested in studying regularity up to the boundary for one-phase singularly perturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely $$ F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = ζ_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad Ω\subset \R^n $$ where $ζ_{\varepsilon}$ behaves asymptotically as the Dirac measure $δ_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.
We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of the elliptic problem for D-N operator. To begin with, we study elliptic problems with mixed boundary condition to derive singularity decompositions and estimates. Based on the analysis, we present the estimates for both D-N operator and its shape derivative with the existence of singular parts.
By means of a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$ recently obtained, we give some necessary conditions for locating the concentration points for a class of quasi-linear elliptic problems in divergence form. More precisely we show that the points where the concentration occurs must be critical, either in a generalized or in the classical sense, for a suitable ground state function.
Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. Also a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.
We present a necessary and sufficient condition on nonnegative Radon measures $μ$ and $ν$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-Δu=μu^q+ν$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.
Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying Hörmander's condition and ${Δ_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ${Δ_L}$ and $Δ_L^2$. These inequalities extend Yang's inequalities for Dirichlet eigenvalues of Laplacian to the settings here and the forms of inequalities are more general than Yang's inequalities. To obtain them, we give a generalization of the inequality by Chebyshev.
We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.
This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, …
In this paper we present a very simple proof of the existence of at least one non trivial solution for a Kirchhoff type equation on $\RN$, for $N\ge 3$. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap betwe…
Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $(q_{11},\dots, q_{nn})\in ]0,+\infty[^n$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_{p,ε}]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,\dots,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a pa…
We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the Perron-Wiener-Brelot solutions of the problem. To this end, we extend the usual notion of the trace operator to Sobolev space $H^1(D)$ with $D$ being an arbitrary bounded open subset of $\mathbb{R}^d$. In the second part of the paper, we prove some existence results for the Dirichlet problem for semilinear equations with measure data on the right-hand side and $L^1$-da…
In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as $p$-{\it ellipticity}. Specifically, let $Ω$ be a chord-arc domain in $\mathbb R^n$ and the operator $\mathcal L = \partial_{i}\left(A_{ij}(x)\partial_{j}\right) +B_{i}(x)\partial_{i} $ be elliptic, with $|B_i(x)| \le Kδ(x)^{-1}$ for a small $K$. Let $p_0 = \sup\{p>1: A \,\,\text{is}\,\, \text{$p$-elliptic}\}$. We establish that if the $L^q$ Dirichlet problem is solvable for $\mathcal L$ for some $1<q< \frac{p_0(n-1)}{(n-2)}$, then the $L^p$ Dir…
Based on the $H^2$ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $\om\in(0,π/2)$. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $\om\in(0,π)$.
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.
In this paper we prove Holder regularity of the gradient for solutions of Dirichlet problem associate to degenerate elliptic equations, extending the recent result of Imbert and Silvestre. Indeed we obtain regularity up to the boundary and when the equation has lower order terms.The proof follows their scheme but requires new tools and new ideas. In particular we give some a priori Lipschitz and Hölder estimates in the presence of boundary condition on one part of the boundary.
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.
The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have been used. Since, found Green's function contains a hypergeometric function of Appell, we had to deal with decomposition formulas, formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem.
This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some function $\bar u(x)$ uniformly on every compact subset $K$ of the domain $D$. Moreover, $\bar u$ is a solution to some boundary value problem. For this result, we assume that the boundary of the domain has no (rational) flat spots and the ratio of elliptic constants $Λ/ λ$ is sufficiently large.
Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon>0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\varepsilon$. A sharp order operator error estimate $\|\mathcal{A}_{D,\varepsilon}^{-1} - (\mathcal{A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon$ is obtained. Here $\mathcal{A}^0_D$ is the effective operator with con…
