We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-Δ_p u-\fracμ{|x|^p}u^{p-1}=\frac{C}{|x|^σ}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $μ\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,σ)\in\R^2$ such that the equation has no positive (super) solutions.
The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prüfer-Transformation.
We consider an inverse problem of determining the coefficients of a fractional $p\,$-Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficents. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.
We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp con…
We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,γ}$ with $γ\in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to…
This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_β}}{|x|^β} \mathrm{d}x\right)^\frac{p}{p^*_β},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_β>0$, where $1<p<N$, $0\leq β<p$, $p^*_β=\frac{p(N-β)}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $|x|^{-β}$, some new estimates are established. Based on …
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore--Gibson--Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition.
Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities to establish nonexistence of extremizers for and improve known sharp Hardy inequalities. Secondly, we establish geometrically interesting remainders for the Davies-Hardy-Tidblom inequalities for the mean distance function, as well as generalize and improve several Hardy type inequalities in the spirit of Brezis and Marcus and spectral estimates of Davies. L…
This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem. Applications of these estimates to bilinear type commutator estimates for Laplace operator with Dirichlet boundary condition in exterior domain are discussed too.
We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.
We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in $\mathbb R^N$ $$-Δ_p u -\fracγ{|x|^p} u^{p-1}=u^{p^*-1}$$ where $1<p<N$, $0\le γ<\left(\frac{N-p}{p}\right)^p$ and $p^*=\frac{Np}{N-p}$. The main result is to show that all the solutions in $\mathcal D^{1, p}(\mathbb R^N)$ are radial and radially decreasing about the origin.
We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami …
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.
In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains.
In this paper, we study Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and related the harmonic function theory introduced by Muckenhoupt--Stein. We establish the Moser type inequality for these harmonic functions, which is missing in this setting before. We then apply it to give a direct proof for the equivalence of characterizations of the Hardy spaces associated to Bessel operator via non-tangential maximal function and radial maximal function defined in terms of the Poisson semigroup.
We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $Σ$ intersected with the exterior of a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $Σ\capΩ$ must be the intersection of the Wulff shape and $Σ$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.
We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval $(0,1)$ associated with the fractional Laplace operator $(-\partial_x^2)^s$, where $0<s<1$. We show that there is a control function which is localized in a non-empty open set $\mathcal{O}\subset \left(\mathbb{R}\setminus(0,1)\right)$, that is, at the exterior of the interval $(0,1)$, such that the system is null controllable at any time $T>0$ if and only if $\frac 12<s<1$.
We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace eq…
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
In this paper, we establish a general weighted Hardy type inequality for the $% p-$Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls with radius $R$ at the origin in $\mathbb{R}^n$.
Existence, non-existence and degeneracy of limit solutions to $p$-Laplace problems involving Hardy potentials as $p\to1^+$. The case of a critical drift
In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: \begin{equation}\label{GPDE0} (-Δ)^{\fracα{2}}u(x)=f(x,u) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\,\, Ω_{r}:=\{x\in\mathbb{R}^{n}\,|\,|x|>r\} \end{equation} with arbitrary $r>0$, where $n\geq2$, $0<α\leq 2$ and $f(x,u)$ satisfies some assumptions. A typical case is the Hardy-Hénon type equations in exterior domains. We first derive the equivalence between \eqref{GPDE0} and the corresponding integral equations \begin{equation}\label{GIE0} u(x)=\int_{Ω_{r}}G_α(x,y)f(y,u(y)…
We study the existence and nonexistence of singular solutions to the equation $u_t-Δu - \fracκ{|x|^2}u+|x|^αu|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that $\a>-2$ and $κ<\left(\frac{N-2}2\right)^2$. The problem is transferred to the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a {\it source solution} with a multiple …
This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar wave front as time goes to negative infinity. In particular, some disturbances on the profile of the entire solution happen as the entire solution comes to the interior domain. But the disturbances disappear as the entire solution is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile and cont…
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered $L^2$ Hardy-Sobolev inequality.
Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-Δ_p u-μ|x|^{-p}{u^{p-1}}=|x|^{-s}{u^{\crits-1}}+u^{\crit-1}$ admits a positive weak solution in $\rn$ of class $\dunp\cap C^1(\rn\setminus\{0\})$, whenever $μ<μ_1$, and $μ_1=[(n-p)/p]^p$. The technique is based on the existence of extremals of some Hardy--Sobolev type embeddings of independent interest. We also show that if $u\in\dunp$ is a weak solution in $\rn$ of $-Δ_p u-μ|x|^{-p}{|u|^{p-2}u}=|x|^{-s}{|u|^{\crits-2}u}+|u|^{q-2}u$, then $u\equiv0$ when either $1<q<\crit$, or $q>\crit$ and $u$ is also of class $L^\i…
We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. In the first part, we show that if $0<s<1$, $Ω\subset\RR^N$ ($N\ge 1$) is a bounded Lipschitz domain and the parameter $δ> 0$, then there is no control function $g$ such that the following system \begin{equation*} \begin{cases} u_{tt} + (-Δ)^{s} u + δ(-Δ)^{s} u_{t}=0 & \mbox{ in }\; Ω\times(0,T),\\ u=gχ_{\mathcal O\times (0,T)} &\mbox{ in }\; (\Omc)\times (0,T) ,\\ u(\…
Starting with a functional difficulty appeared in the paper \cite{vz00} by Vázquez and Zuazua, we obtain new insights into the Hardy Inequality and the evolution problem associated to it by means of a reformulation of the problem. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem arises when the equation is posed in a bounded domain, and also when posed in the whole space. We also consider an equivalent problem with in…
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula and a Green's formula for hyperbolic harmonic functions. Using a Cauchy Integral formula we are able to introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic fu…
Let $\Om\subset\RR^N$ a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusion equation \begin{equation*} \begin{cases} \mathbb D_t^αu+(-Δ)^su=0\;\;&\mbox{ in }\;(0,T)\timesΩ\\ u=g &\mbox{ in }\;(0,T)\times(\RR^N\setminusΩ)\\ u(0,\cdot)=u_0&\mbox{ in }\;Ω, \end{cases} \end{equation*} where $u=u(t,x)$ is the state to be controlled and $g=g(t,x)$ is the control function which is localized in a subset $\mathcal O$ of $\Omc$. Here, $0<α\le 1$, $0<s<1$ and $T>0$ be real numbers. After giving an explicit representation of solutions, w…
Let $0<σ<n/2$ and $H=(-Δ)^σ+V(x)$ be Schrödinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2σ}$ with a sharp coupling constant $a\ge -C_{σ,n}$ ($C_{σ,n}$ is the best constant of Hardy's inequality of order $σ$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all …
We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $ϕ_h$, $(h^2 Δ- 1)ϕ_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^δ$-scales ($0\leq δ< 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrödinger operators.
We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary …
