Gianpaolo Piscitelli
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.
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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.
In this paper we introduce a new type of symmetrization, which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function, in order to prove sharp comparison results for solutions of a class of homogeneous Dirichlet fully nonlinear elliptic problems of second order and for suitable anisotropic Hessian integrals.
We prove estimates for weak solutions to a class of Dirichlet problems associated to anisotropic elliptic equations with a zero order term.
In this paper we introduce new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show a Pólya-Szego type principle for such symmetrization -- it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotrop…
In this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result.
We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e., integral) comparison for fractional elliptic equations of the type $(-Δ)^{s}u=f$ $(0<s<1)$ in a bounded domain $Ω$, equipped with homogeneous boundary conditions. The classical pointwise Talenti rearrangement inequality is recovered in the limit $s\rightarrow1$. Finally, explicit counterexamples constructed for all $s\in(0,1)$ highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.
We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first $p$-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain $L^{\infty}$ estimates for solutions of nonlinear and eigenvalue problems in terms of other $L^p$ norms.
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga-Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates i…
In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain $Ω$ of $\R^n$ and the solutions of some radially symmetric equations in the equimeasurable ball $Ω^*$. The coefficients of the symmetrized equations in~$Ω^*$ satisfy similar constraints as the original ones in~$Ω$. We consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient. Lastly, we show some improved quantified comparisons when the original domain is not a ball. Th…
We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $Ω$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $Ω$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational…
The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization
本文证明各向异性高斯等周不等式并刻画取等条件，同时通过构造反例说明Ehrhard对称化不满足递减性，进而提出含误差项的新不等式，为证明各向异性Ehrhard对称化的唯一性提供依据。
We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization.
In this paper we obtain comparison results for the quasilinear equation $-Δ_{p,x} u - u_{yy} = f$ with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable $x$, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in $y$, and the proof of a comparison principle for the discrete version of the auxiliary problem $A U - U_{yy} \le \int_0^s f^*$, where $AU = (nω^{1/n}s^{1/n'} )^p (- U_{ss})^{p-1}$. We show that this operator is T-accretive in $L^\infty$. We extend our resu…
Anisotropic gradient rearrangement of BV functions and applications
本文提出了一种各向异性设定下BV函数分布梯度的对称化技术，通过分离绝对连续部分与奇异部分，推导出函数与其各向异性对称化的L^1比较，并应用于此类几何泛函的等周不等式。
We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-Δ)^{σ/2}u=f$ and itselliptic counterpart $h v +(-Δ)^{σ/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u+(-Δ)^{σ/2}A(u)=f$, but only when $A:\re_+\to\re_+$ is a concave function. In the elliptic case, complete symmetrization results are proved for $\,B(v)+(-Δ)^{σ/2}v=f$ \ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results when $B$ is concave. Remarkable counterexamples…
We solve a class of weighted isoperimetric problems of the form $$ \min\left\{\int\_{\partial E}w e^V\,dx:\int\_E e^V\,dx={\rm constant}\right\} $$ where $w$ and $V$ are suitable functions on $\R^d$. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations.
A method for proving symmetrization inequalities for some elliptic p.d.e.'s on manifolds equipped with appropriate isoperimetric inequalities is outlined. The method is based on a modification of an approach of Baernstein. The question of what is the most general result that can be proved in this way is still open, and the author can be consulted if the reader is interested in this question.
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 develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equation of mean curvature type. We conclude the paper discussing possible applications to the isoperimetric problem in symmetric spaces of noncompact type.
In this article, we prove a sharp estimate for the solutions to parabolic equations on manifolds. Precisely, using symmetrization techniques and isoperimetric inequalities on Riemannian manifold, we obtain a Bandle's comparison on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. Our results generalize Bandle's result [6] to Riemannian setting, and Talenti's comparison for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and Chen-Li [9] to parabolic equations.
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in [18, Theorem 31]. Some interesting consequences are Lp regularity results and nonlocal energy estimates for solutions.
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $Ω$ with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as $Ω$. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.
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.
Global asymptotic expansion for anisotropic singularly perturbed elliptic problems
针对一类线性椭圆问题的各向异性奇异扰动，本文给出并证明了解在特定函数空间中的全局渐近展开。
The minimizers of the anisotropic fractional isoperimetric inequality with respect to the convex body $K$ in $\mathbb{R}^n$ are shown to be equivalent to star bodies whenever $K$ is strictly convex and unconditional. From this a Pólya-Szegö principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.
Liouville-type results for some quasilinear anisotropic elliptic equations
本文证明了拟线性各向异性椭圆方程稳定解的Liouville型定理，并涵盖纯Finsler p-Laplacian情形。
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.
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.
Anisotropic Calderón Problem for a Non-Local Second Order Elliptic Operator
本文研究了闭Riemann流形上二阶非局部椭圆算子的各向异性Calderón问题，证明了利用Cauchy数据集可在标准规范下恢复流形几何结构。
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 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.
Second order regularity for solutions to anisotropic degenerate elliptic equations
本文研究退化各向异性椭圆方程解的正则性，建立了二阶估计并给出应力场正则性结果，且结果在欧几里得情形下亦为新。
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.
Existence and uniqueness of solutions to some anisotropic elliptic equations with a singular convection term
研究具有奇异对流项的各类各向异性椭圆方程，证明了弱解的存在唯一性，并推导了先验估计与正则性结果。
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.
Existence and uniqueness of weak solutions to singular anisotropic elliptic problems
本文研究了一类带p次次线扰动的非线性各向异性椭圆问题，证明了弱解的唯一性，并给出了存在性的必要充分条件。
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
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.
Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities
研究各向异性 p-拉普拉斯拟线性椭圆系统，给出保证非平凡解存在与不存在的条件，涉及临界 Sobolev 增长。
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.
Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
本文通过推广Poincaré型公式，证明了二维和三维空间中拟线性各向异性椭圆方程稳定解的一维对称性结果。
Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth
本文针对p-Laplace型各向异性椭圆算子的Dirichlet和Neumann边值问题，在最弱右端项假设下建立了解的全局Lipschitz正则性，结果适用于凸域或边界曲率满足最小可积性假设的域。
Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights
本文研究具有各向异性权重的非线性椭圆与抛物方程在退化或奇异点附近的渐近行为，并建立解的局部Hölder估计。
On some Elliptic and Parabolic Problems Involving the Anisotropic $p(u)$-Laplacian
研究由各向异性$p(u)$-拉普拉斯算子驱动的椭圆和抛物方程，利用变指数Sobolev空间、伪单调算子、时间离散及Schauder不动点定理，证明了弱解的存在性。
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, 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…
Existence of positive solutions to anisotropic elliptic equations in the entire Euclidean space
本文针对全欧氏空间中非线性项具指数或临界增长的各向异性椭圆方程，运用山路引理及Ekeland变分原理，证明其存在非平凡正弱解，且扰动后有两个不同正弱解。
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 article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet…
We are concerned with singular elliptic problems of the form $-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$ in $Ω,$ where $Ω$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partialΩ),$ $\la>0,$ $μ\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $Ω$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u|^a$, and the singular nonlinearity $g$, w…
We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion of each density is described by a different operator.
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