We prove that a certain class of elliptic free boundary problems, which includes the Prandtl-Batchelor problem from fluid dynamics as a special case, has two distinct nontrivial solutions for large values of a parameter. The first solution is a global minimizer of the energy. The energy functional is nondifferentiable, so standard variational arguments cannot be used directly to obtain a second nontrivial solution. We obtain our second solution as the limit of mountain pass points of a sequence of $C^1$-functionals approximating the energy. We use careful estimates of the corresponding energy levels to show that this limit is neither trivial nor a minimizer.
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.
In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems.
We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lamé system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied.
We give full boundary extensions to two fundamental estimates in the theory of elliptic PDE, the weak Harnack inequality and the quantitative strong maximum principle, for uniformly elliptic equations in non-divergence form.
Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic problem with Neumann boundary conditions .
In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.
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 paper, existence and localization results of $C^1$-solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder.
This paper is concerned with the convergence of the solution of general elliptic boundary value problems in cylindrical domain, when some directions of the domain go to infinity.
We consider an elliptic pseudo differential equation in a multi-dimensional cone and starting wave factorization concept we add some boundary conditions. For the simplest cases explicit formulas for solution are given like layer potentials for classical case.
We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and subcritical superlinear free boundary problems, and establish full regularity of the free boundary in dimension $N = 2$ and partial regularity in higher dimensions.
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 quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to $L^\infty(\close)\times L^\infty(\close)$. The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.
The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its Ricci curvature.
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.
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
In this paper, a geometric condition on domains will be given which guarantees the boundary differentiability of solutions of elliptic equations, that is, the solutions are differentiable at any boundary point. We will show that this geometric condition is optimal.
This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
Under a Morse index condition we prove symmetry results for solutions of a nonlinear mixed boundary condition elliptic problem. As an intermediate step we relate the Morse index of a solution to a mixed boundary condition linear eigenvalue problem for which we construct sequences of eigenvalues and provide variational characterization of them.
We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence result of large solution to a similar problem.
In this paper, we prove the pointwise boundary differentiability for viscosity solutions of fully nonlinear elliptic equations. This generalizes the previous related results for linear equations. The geometrical conditions in this paper are pointwise and more general than before. Moreover, our proofs are relatively simple.
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
We investigate a general elliptic problem given in a bounded Euclidean domain with boundary data in Nikolskii spaces of low, specifically, negative order. The right-hand side of the elliptic differential equation is supposed to be an integrable function. We establish the Fredholm property of the problem, the maximal regularity and a priori estimate of its generalized solutions in the spaces indicated. We give an application of these results to some elliptic problems with boundary conditions induced by a Gaussian white noise.
We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.
We study geometric and regularity properties of the largest subsolution of a one-phase free boundary problem under a very general free boundary condition in R2. Moreover, we provide density bounds for the positivity set and its complement near the free boundary.
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is $C^{1,α}$ on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is $C^{2,α}$ on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.
In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.
Given a compact manifold with boundary endowed with an isometric action of a discrete group of polynomial growth, we state an index theorem for elliptic elements in the algebra of nonlocal operators generated by the Boutet de Monvel algebra of pseudodifferential boundary value problems on the manifold and the shift operators associated with the group action.
We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free boundaries are $C^{1, γ}$.
Comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated in the last decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved.
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.
In this paper, we establish $C^{1, α}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, α}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the non-existence regimes. For the existence results, we make use of a particular class of weighted Sobolev spaces to compensate boundary singularities which are naturally built in the problem.
We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.
We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. From this, elliptic and parabolic regularity results are deduced by means of Sneiberg's isomorphism theorem.
Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
We show that two classically known properties of positive supersolutions of uniformly elliptic PDEs, the boundary point principle (Hopf lemma) and global integrability, can be quantified with respect to each other. We obtain an extension to the boundary of the de Giorgi-Moser weak Harnack inequality, optimal with respect to the norms involved, for equations in divergence form.
This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<\infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $α$ for any $α\in (0,1)$.
We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.
We consider a function U satisfying a degenerate elliptic equation on (0,+\infty)\times R^N with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain Ω\subset R^N of class C^{1;1}, whereas the Dirichlet data is on the exterior of Ω. We prove Holder regularity estimates of U/d^s, where d is a distance function defined as d(z) := dist(z;R^N\setminusΩ), for z\in (0,+\infty)\times R^N. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness…
We consider the Dirichlet problem for quasilinear elliptic equations with Musielak-Orlicz (p,q)-growth and non-logarithmic conditions on the coefficients. A sufficient Wiener-type condition for the regularity of a boundary point is established.
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.
The dependence of the smoothness of variational solutions to the first boundary value problems for second order elliptic operators are studied. The results use Sobolev-Slobodetskii and Nikolskii-Besov spaces and their properties. Methods are based on real interpolation technique and generalization of Savaré-Nirenberg difference quotient technique.
