In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $Ω\subset \mathbb{R}^N$ whose boundary $\partialΩ$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to $u$ the convolution $ρ\ast E(u)$ of $ρ$ with $E(u)$, where $ρ$ is an integrable function on $\mathbb{R}^N$ and $E$ is an extension operator related to $Ω$. Under verifiable conditions, we prove the existence of a (weak) solution to our problem by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to our problem is bounded.
In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal Carleson measure estimate for bounded solutions of such operators implies a regularity result for the associated elliptic measure.
In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
In this paper we utilise new methods of Calculus of Variations in $L^\infty$ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in $L^2$ is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.
We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation $-Δu=f(u)$ in the whole $R^N$, where $f\in C^1(R)$ is a general nonlinearity and $N\geq 1$ is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.
We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint $BMO$ Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the $BMO$ Dirichlet problem implies $L^p$ solvability for all $p>p_0$.
We prove L^$\infty$ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial L^$\infty$ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian sol…
In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov re…
In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $Ω$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where the growth condition for the function $B\colon Ω\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.
We show that a second-order elliptic differential operator $P$, on any manifold $M$, has closed range in $C^\infty(M)$. If $M$ has no compact components, then $P$ is surjective on $C^\infty(M)$. Applications to Helmholtz decomposition are discussed.
In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $Ω\subset \mathbb{R}^N$, we show that a solution $ϕ\in W_0^{1,\infty}(Ω)$ to an appropriate elliptic equation $\mathcal{L} ϕ= F$, with $F \in L^{\infty}(Ω;\mathbb{R})$, satisfies $|\nabla ϕ|_{\infty} \leq C |F|_{\infty}$, with a positive constant $C = \exp(C(\mathcal{L})\text{diam}(Ω))$. We also obtain similiar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series …
We consider layer potentials associated to elliptic operators $Lu=-{\rm div}(A \nabla u)$ acting in the upper half-space $\mathbb{R}^{n+1}_+$ for $n\geq 2$, or more generally, in a Lipschitz graph domain, where the coefficient matrix $A$ is $L^\infty$ and $t$-independent, and solutions of $Lu=0$ satisfy interior estimates of De Giorgi/Nash/Moser type. A "Calderón-Zygmund" theory is developed for the boundedness of layer potentials, whereby sharp $L^p$ and endpoint space bounds are deduced from $L^2$ bounds. Appropriate versions of the classical "jump-relation" formulae are also derived. The me…
In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}λ_{1,p}=λ_{1,\infty}=\left(\fracπ{2R}\right)^2$ where $λ_{1,p}$ and $λ_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.
We consider the semilinear elliptic equation $-Δu =λf(u)$ in a smooth bounded domain $Ω$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $Ω$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabré \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<β_{-}:=\li…
We consider the semilinear elliptic equation $-L u = f(u)$ in a general smooth bounded domain $Ω\subset R^{n}$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. We prove that if $u$ is a positive semistable solution then for every $0\leqβ<1$ we have $$f(u)\int_{0}^{u}f(t)f"(t)~e^{2β\int_{0}^{t}\sqrt{\frac{f"(s)}{f(s)}}ds}~dt\in L^{1}(Ω),$$ by a constant independent of $u$. As we shall see, a large number of result…
The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-Δu =\frac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{α, μ}^{\ast}}}{|x-y|^μ|y|^α}dy\right) |u|^{2_{α, μ}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, …
We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(ϕ(|\nabla u|)\nabla u\big)+V(x)ϕ(|u|)u=f(u)\qquad u\in W^{1,Φ}(\mathbb{R}^{N}), \] where $ϕ$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,Φ}(\mathbb{R}^{N})$ into $L^Φ(\mathbb{R}^{N})$ is established, whic…
We prove that if the elliptic problem $-Δu+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $Ω$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_Ωcϕ^2}\le \sqrt{ \int_Ω| \nablaϕ|^2}+\sqrt{ \int_Ω\frac{b^2}{4}ϕ^2},~~~ϕ\in C_c^\infty(Ω).\] As an application, we obtain Liouville type theorems for positive supersolutions in exterior domains when $c(x)-\frac{b^2(x)}{4}>0$ for large $|x|$, but unlike the known results we allow the case $\liminf_{|x|\rightarrow\infty}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$ and $c$ are allowed to be unbounded. …
Let $Ω$ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{Ω}) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $Ω$ and $u(x)=0, \ |\nabla u (x)| =λ(|x|)$ on $\partial Ω$, where $g\in C([0,+\infty ))\cap C^1 ((0,+\infty ) ) $ with $g(0)=0$, $g'(t)>0$ for $t>0$, $f\in C([0,+\infty )) \times [0, +\infty ) )$, $f$ is nonincreasing in $|x|$, $λ\in C([0, +\infty )) $ and $λ$ is positive and nondecreasing. We show that $Ω$ is a ball and $u$ satisfies some "local" kind of symmetry. The proof is based o…
We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -Δu =λf(v) \quad \mbox{ in } Ω -Δv =γf(u) \quad \mbox{ in } Ω, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $λ,γ$ are positive parameters, $Ω$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. Assuming $$0<τ_{-}:=\liminf_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq 2…
For a Hamiltonian $H \in C^2(\mathbb{R}^{N \times n})$ and a map $u:Ω\subseteq \mathbb{R}^n /!\longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|H(Du)\big\|_{L^\infty(Ω)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} \tag{2} A_\infty u := \Big(H_P \otimes H_P + H[H_P]^\bot /! H_{PP}\Big)(Du):D^2 u = 0. \] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [K1]. Herein we show that the D…
We study the inequality $ -Δu - \fracμ{|x|^2} u \geq (|x|^{-α} * u^p)u^q$ in an unbounded cone $\mathcal{C}_Ω^ρ\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $Ω$ of the unit sphere $S^{N-1}\subset \mathbb{R}^N,$ $p, q, ρ>0$, $μ\in \mathbb{R}$ and $0\leq α< N$. In the above, $|x|^{-α} * u^p$ denotes the standard convolution operator in the cone $\mathcal{C}_Ω^ρ$. We discuss the existence and nonexistence of positive solutions in terms of $N, p, q, α, μ$ and $Ω$. Extensions to systems of inequalities are also investigated.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.
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.
In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations invo…
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …
In this paper, we study the following elliptic problem with Dirac mass \begin{equation}\label{eq 0.1} -Δu=Vu^p+k δ_0\quad {\rm in}\quad \mathbb{R}^N, \qquad \lim_{|x|\to+\infty}u(x)=0, \end{equation} where $N>2$, $p>0$, $k>0$, $δ_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\mathbb{R}^N\setminus\{0\}$ satisfying $$ V(x)\le \frac{c_1}{|x|^{a_0}(1+|x|^{a_\infty-a_0})} $$ with $a_0<N,\ a_\infty>a_0 $ and $c_1>0$. We obtain two positive solutions of (\ref{eq 0.1}) with additional conditions for parameters on $a_\infty, a_0$, $p$ and $k$. The first solution …
We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -Δ(u γ(v)= μu(1-u), \\ - Δv +v=u, \end{array} \right. $$ under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $Ω$ of $\R^N$ (for $N \geq 1)$, where $γ\in C^3([0, \infty))$ and satisfies the assumptions $γ(s) > 0$, $γ^{\prime}(s) \leq 0$, $γ^{\prime \prime} (s) \geq 0$, $γ^{\prime \prime \prime}(s) \leq 0$ for any $s \geq 0$ $$-2 γ^{\prime}(s) + γ^{\prime \prime}(s)s \leq μ_0< μ$$ $$\frac{[γ^{\…
In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $Γ$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher condition of approximation by constant coefficient matrices, with a small enough Carleson norm, and show that the elliptic measure with pole at infinity associated to $L$ is $A_\infty$-absolutely continuous with respect to the surface measure on $Γ$, with a small $A_\infty$ constant. In other words, we show that for relatively flat uniformly rectifiable sets and for …
In this paper we study, in an open bounded set $Ω\subset\mathbb R^N$ with Lipschitz boundary $\partialΩ$, the Dirichlet problem for a nonlinear singular elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation appearing in the level set formulation of the inverse mean curvature flow. Our aim is twofold. On the one hand, we consider data belonging to the Marcinkiewicz space $L^{N,\infty}(Ω)$, which leads to unbounded solutions. So, we have to begin introducing the suitable notion of unbounded solution to this problem. Moreover, e…
This paper is focused on the local interior $W^{1,\infty}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u, \nabla u)] +b(x, u, \nabla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^\infty$-norm for the solution's gradient in terms of its local $L^p$-norm. Specifically, we prove \begin{equation*} \|\nabla u\|_{L^\infty(B_{\frac{R}{2}}(x_0))}^p \leq \frac{C}{|B_R(x_0)|}\int_{B_R(x_0)}|\nabla u(x)|^p dx. \end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{…
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $ω_L\in A_{\infty}(σ)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.
We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain $Ω$ with Ahlfors regular boundary, and we show that the $A_\infty$ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher.
Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient matrix satisfies $(C(x) \, ξ, ξ) \in Σ_θ$ for all $x \in \mathbb{R}^d$ and $ξ\in \mathbb{C}^d$, where $Σ_θ$ be the sector with vertex 0 and semi-angle $θ$ in the complex plane. We show that for all $p$ in a suitable interval the contraction semigroup generated by $-A$ extends consistently to a contraction semigroup on $L_p(\mathbb{R}^d)$. For those values of $p$ w…
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
In this work, we present new results on solvability of the equation $A^{*}(D)f=μ$ for $f \in L^{p}$ and positive measure data $μ$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $μ$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.
The first aim of this paper is to develop a layer potential theory in $L_2$-based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ${\mathbb R}^n$, $n\geq 3$, for the anisotropic Stokes system with $L_{\infty }$ viscosity coefficient tensor satisfying an ellipticity condition for symmetric matrices. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ${\mathbb R}^n$, with the given data in $L_2$-based weighted Sobolev spaces. These results are us…
Let $n\geq 3$, $α$, $β\in\mathbb{R}$, and let $v$ be a solution $Δv+αe^v+βx\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}ρ^{1-n} (\int_{B_ρ}e^v\,dx)dρ\in (0,\infty)$ and $|x|^2e^{v(x)}\le A_1$ in $\R^n$. We prove that $\frac{v(x)}{\log |x|}\to -2$ as $|x|\to\infty$ and $α>2β$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.
The paper concerns with positive solutions of problems of the type $-Δu+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $Ω\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $Ω$ can be an exterior domain, i.e. $\mathbb{R}^N\setminusΩ$ bounded, or the whole of $\mathbb{R}^N$. The potential $a\in L^{N/2}_{\rm loc}(\mathbb{R}^N)$ is assumed to be strictly positive and such that there exists $\lim_{|x|\to\infty}a(x):=a_\infty$, with $a_\infty>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge …
Using Harnack's inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in \partialΩ\cup\{\infty\}$ for the quasilinear elliptic equation $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{ in } Ω,$$ where $Ω$ is a domain in $\mathbb{R}^d$, $d\geq 2$, $1<p<d$, and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω; \mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. It is assumed that the potential $V$ belongs to a certain Wolff class and has a generalized Fu…
Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.
We consider the Dirichlet problem for positive solutions of the equation $-Δ_p (u) = f(u)$ in a convex, bounded, smooth domain $Ω\subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty} $ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty} $ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.
We obtain well-posedness results in $L_p$-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor, in complementary Lipschitz domains of ${\mathbb R}^n$, $n\ge 3$. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in $L_p$-based weighted Sobolev and Besov spaces. We show that such a mixed varia…
Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $Hφ=-(c φ')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $ν=ν_+\veeν_-$ where $ν_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $ν\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $ν\not\in L_\infty(0,1)$. I…
This paper is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the contrast ratio $δ^2$ for $0<δ<\infty$. Our study also covers the case of soft inclusions ($δ=0$) as well as the case of stiff inclusions ($δ=\infty$). The large-scale Lipschitz estimate, together with classical local estimates, allows us to establish explicit bounds for the matrix of fundamental solutions and its derivatives.
In this note, we explore some consequences of the Modica-Mortola construction of a singular elliptic measure, as regards the link between the quantitative absolute continuity ($A_{\infty}$) of their approximations and the suitability of a well-known tool, the so-called Kenig-Pipher condition ($\operatorname{KP}$). The Kenig-Pipher condition is used to ascertain absolute continuity in the presence of some mild regularity of the coefficient matrix. We perform some modifications of the Modica-Mortola example to show the following two statements: (a) There are sequences of matrices for which both …
In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^α)Δ+ b|x|^{α-2}x\cdot\nabla - c |x|^{α-2} - |x|^β , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $α\in[0,2)$, $β>0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely t…
On a manifold $X$ with boundary and bounded geometry we consider a strongly elliptic second order operator $A$ together with a degenerate boundary operator $T$ of the form $T=φ_0γ_0 + φ_1γ_1$. Here $γ_0$ and $γ_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $φ_0,φ_1\in C^{\infty}_b(\partial X)$, $φ_0,φ_1\ge 0$, and $φ_0+φ_1\geq c$, for some $c>0$. We also assume that the highest order coefficients of $A$ belong to $C^τ(X)$ for some $τ>0$ and the lower order coefficients are in $L_\infty(X)$. We show that the $L_p(X)$-rea…
We obtain an a-priori $W_{loc}^{1,\infty }\ ( Ω;\mathbb{R}^{m}\ ) -$bound for solutions in $Ω\subset \mathbb{R}^{n} $, $n\geq 2$, to the elliptic system \begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}\ ( \frac{g_{t}\ ( x,\ |Du\ | \ ) }{\ |Du\ | } u_{x_{i}}^{α}\ ) =0,\;\;\;\;\;α=1,2,\ldots ,m, \end{equation*} where $g\ ( x,t\ ) $, $g:Ω\times \ [ 0,\infty \ ) \rightarrow \ [ 0,\infty \ ) $, is a Carathéodory function, convex and increasing with respect to the gradient variable $t\in \ [ 0,\infty \ ) $. We allow $x-$dependence, which turns out to be a relevant difference with res…
