Using the sub-supersolution method we study the existence of positive solutions for the anisotropic problem \begin{equation} -\sum_{i=1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right)=λu^{q-1} \end{equation} where $Ω$ is a bounded and regular domain of $\mathbb{R}^N$, $q>1$ and $λ>0$.
In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.
We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter $1/\varepsilon$ in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a…
In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.
We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
In this note we establish the large time non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.
We study exponential decay rates of eigenfunctions of self-adjoint higher order elliptic operators on R^n. We are interested in decay rates as a function of direction. We show that the possible decay rates are to a large extent determined algebraically.
In this article, we study, as the coefficient $s\to\infty$, the asymptotic behavior of the principal eigenvalue of $$-φ''(x)-2sm'(x)φ'(x)+c(x)φ(x)=λ_sφ(x),\ \ 0<x<1,$$ supplemented by different boundary conditions. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term $m$ allows natural degeneracy. For instance, $m$ could be constant on $[a,b]\subset[0,1]$. Depending on the behavior of $m$ near the neighbourhood of the end points $a,\,b$, the limiting value could be the principal eigenvalue of $$-φ''(x)…
Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators
School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；Academy of Mathematics and Systems Science, Academia Sinica（中国科学院数学与系统科学研究院）
We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.
We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.
Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schrödinger operators and control theory. We review recent results and announce new ones regarding quantitative unique continuation principles for partial differential equations with an underlying multiscale structure. They concern Schrödinger and second order elliptic operators. An important feature is that the estimates are scale free and with quantitative dependence on parameters. These unique continuation principles apply to functions satisfying certain `r…
For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O(ε)$ estimate in $H^1$ for solutions with Dirichlet condition.
Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained using symmetrization techniques.
We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $Ω\subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge δ> 0$ for all $x \in \overlineΩ$, even if the boundary $\partial Ω$ is only Lipschitz continuous. Under such weak regularity assumptions the Hopf-Ole\uınik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map $L_p(Ω)$ into $C(\overlineΩ)$. The same tool also yields corresponding results for Dirichlet or mixed boundar…
We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic op…
In this paper, we deal with anisotropic singular perturbations of some class of elliptic problem. We study the asymptotic behavior of the solution in certain second order pseudo Sobolev space.
In this paper we prove some Harnack inequality for fully non linear degenerate elliptic equations, in the two dimensional case, extending the results of Davila Felmer and Quaas in the singular case but in all dimensions. We then apply this result for the existence of an eigenfunction in smooth bounded domain.
We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.
In this article we study the asymptotic behavior, of the solution of a nonlinear elliptic, anisotropic singular perturbations problem in cylindrical domain, the limit problem is given and strong convergences are proved, we also give an application to intergo-differential problems.
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $ϕ(Ω)$ parametrized by Lipschitz homeomorphisms $ϕ$ defined on a fixed reference domain $Ω$. Given two open sets $ϕ(Ω)$, $\tilde ϕ(Ω)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde ϕ-ϕ\|_{W^{1,p}(Ω)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschit…
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a lower-order term depending on the gradient.
We extend the estimates proved by Donnelly and Fefferman and by Lebeau and Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in the control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.
We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $Ω\subset \>I\!\!R^{N}\>(N \geq 2)\>$. We prove the existence of entropy solutions avoiding sign condition and coercivity on the lowers order terms.
We use "generalized" version of total variation, coarea formulas, isoperimetric inequalities to obtain sharp estimates for solutions (and for their gradients) to anisotropic elliptic equations with a lower order term, comparing them with the solutions to the convex symmetrized ones.
The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established.It is proved that the obtained entropy solution is a renormalized solution of the considered problem.
In this paper we study a class of anisotropic equations with a lower order term whose coefficients lay in Marcinkiewicz spaces. We prove some regularity results for local solutions requiring any control on the norm of the coefficients.
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.
In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.
In this article we give an extention of the L^2-theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving L^p data (1<p<2). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains
We discuss properties of $L^2$-eigenfunctions of Schrödinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.
In this paper we study the main properties of the first eigenvalue and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality.
In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behavior of the semigroup generated.
We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak--Orlicz spaces generated by an $N$-function $M:Ω\times\mathbb{R}^d\to[0,\infty)$. Neither $\nabla_2$ nor $Δ_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.
We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity, and some integral and pointwise Harnack inequalities.
A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane.
The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+δ}$ for some $δ>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and Hölder estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x) &\text{in }Ω,\\ u=0&\text{on }\partialΩ. \end{array} \right. $ Here $Ω$ is a bounded open set of $\mathbb R^{N}$, $N\ge 2$, $0<p-1<q\le p<N$, and $\mathcal Q_{p}$ is the anisotropic operator $ \mathcal Q_{p} u ={\rm div}\left( [H(Du)]^{p-1}H_ξ(Du) \right)$, where $H$ is a suitable norm of $\mathbb R^{N}$. Moreover, $f$ belongs to an appropriate Marcinkiewicz sp…
We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $Δ_2$ nor the $\nabla _2$-condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solut…
In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2α}\nabla u)=K(x)|x_{N}|^{α\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $α>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2α}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation.…
In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use st…
In this paper we introduce and study a new Kelvin-type transform in the anisotropic setting. In particular, we deal with semilinear and quasilinear anisotropic elliptic problems in the entire space.
