We consider degenerated nonlinear PDE of elliptic type: $$
- \mathrm{div}(a(|x|)|\nabla w(x)|^{p-2} \nabla w(x)) + h(|x|,w(x),\langle\nabla w(x),\frac{x}{|x|}\rangle)=ϕ(w(x)), $$ where $x$ belongs to the ball in $\bf{R}^n$. Using the argument based on Opial-type inequalities, we investigate qualitative properties of their radial solutions, like e.g. maximum principles, monotonicity, as well as nonexistence of the nontrivial solutions.
We prove some Pólya-Szegö type inequalities which involve couples of functions and their rearrangements. Our inequalities reduce to the classical Pólya-Szegö principle when the two functions coincide. As an application, we give a different proof of a comparison result for solutions to Dirichlet boundary value problems for Laplacian equations proved by A. Alvino, G. Trombetti, J. I. Diaz and P. L. Lions.
A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimisers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The app…
The goal of this note is to have a systematic approach to generating isoperimetric inequalities from two concrete type of PDEs. We call these PDEs Bellman type because a totally analogous equations happen to rule many sharp estimates for singular integrals in harmonic analysis, and such estimates were obtained with the use of Hamilton--Jacobi--Bellman PDE. We show how classical inequalities of Brascamp--Lieb, Prekopa--Leindler, Ehrhard are particular case of this scheme, which allows us to augment the stock of such inequalities. We approach the isoperimetric inequalities as a maximum (minimum)…
We describe a new method of proving a priori bounds for positive supersolutions and solutions of superlinear elliptic PDE, based on global weak Harnack inequalities and a quantitative Hopf lemma. Novel results based on the method include: (i) equations without a boundary condition on the whole boundary; (ii) equations with nonlinearities which do not have precise growth at infinity; (iii) systems of inequalities with opposite sign.
Differential Harnack inequalities and maximum principles of Morel-Oswald type for elliptic PDE in divergence form
PUC-Rio, Departamento de Matematica（天主教里约热内卢大学数学系）；Université Sorbonne Paris Nord, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications（索邦巴黎北大学，法国国家科学研究中心UMR 7539，分析与几何应用实验室）
Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualita…
We prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the ``Soap Bubble Theorem'' for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.
We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincaré inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Hölder continuity in time of order $(p-1)/p$ for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.
We proove some inequalities concerning the product, sup * inf for some elliptic operators of order 2 and 4. Using those inequalities and the concentration phenomena we can describe the asymptotic behavior of those PDE solutions.
We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for the solutions is proved through an argument which uses a boundary growth lemma, and consequences such as boundary "half"-Harnack inequalities, which are of independent interest. Our results are new even for linear equations.
In this article, we prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for "generic" $L^2$ data. When the zero set of a solution (e.g. a harmonic function) contains a singularity, this means that we can find an arbitrarily small perturbation of the boundary data so that the zero set of the perturbed solution is smooth throughout a prescribed neighborhood of the former singularity. Furthermore, we can take the perturbation to be "mean zero" for which there are additional technical difficulties to en…
In this work we prove sharp $L^p$ versions of multipolar Hardy inequalities in the case of a bipolar potential and $p\geq 2$, which were first developed in the case $p=2$ by Cazacu (CCM 2016) and Cazacu&Zuazua (Studies in phase space analysis with applications to PDEs, 2013). Our results are sharp and minimizers do exist in the energy space. New features appear when $p>2$ compared to the linear case $p=2$ at the level of criticality of the p-Laplacian $-Δ_p$ perturbed by a singular Hardy bipolar potential.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
We study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in ${\mathrm{L}}^\infty$, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in ${\mathrm{L}}^p$ as $p\to\infty$ and the generalised Kuhn-Tucker theory to derive the rel…
We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work `…
We develop criteria based on a calibration argument via discrete PDE and semidiscrete optimal transport, for finding sharp isoperimetric inequalities of the form $(\sharp Ω)^{d-1} \le C (\sharp \overrightarrow{\partialΩ})^d$ where $Ω$ is a subset of vertices of a graph and $\overrightarrow{\partialΩ}$ is the oriented edge-boundary of $Ω$, as well as the optimum isoperimetric shapes $Ω$. The method is a discrete counterpart to Optimal Transport and ABP method proofs valid in the continuum, and answers a question appearing in Hamamuki \cite{hamamuki}, extending that work valid for rectangular gr…
This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality. Our approach relies in a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works. We a…
We consider possibly degenerate parabolic operators in the form $$ \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated to a suitable family of stochastic differential equations, and satisfying the Hörmander condition. Note that, under this assumption, the operators in the form $Ł$ has a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods …
In this paper we generalize the classical Strichartz estimation for solutions of initial problem for linear parabolic and Schrödinger PDE on many popular classes {\it pairs} of rearrangement invariant(r.i.) spaces and construct some examples in order to show the exactness of our estimations.
This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $Ω$. Second, we provide a new proof of entire Liouville properties for inequalities with superli…
Global existence for a system of nonlinear partial differential equations (PDE) modeling an isotropic incompressible viscoelastic material is proved. The structure of the PDE is derived through constitutive assumptions on the material. Restriction on the size of the initial displacement and velocity for the model is specified independent of the size of the viscosity of the material. The proof of global existence combines use of vector fields, local energy decay estimates, generalized Sobolev inequalities, and hyperbolic energy estimates.
We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [6] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [11]. Orientable nonlinearities and compatible cost…
We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,α}$ surfaces with smooth and compact boundary.
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of $L_p$ Lyapunov inequalities ($1\leq p\leq \infty$) for differential equations. In this work we review some recent results on these kinds of quest…
Sharp gradient bounds for uniformly elliptic PDE, eigenvalue asymptotics, and the Landis conjecture
PUC-Rio, Departamento de Matematica（天主教巴西里约热内卢大学数学系）；Université Sorbonne Paris Nord, CNRS UMR 7539, Laboratoire Analyse, Géométrie et Applications（索邦巴黎北大学、法国国家科学研究中心7539联合研究单位分析几何与应用实验室）
We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.\,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szegö inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $\mathbb{R}^n$…
Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of Lévy-Gromov and Bérard-Besson-Gallot, as well as some new comparison results.
We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}ν\Vert_{L^q(Ω, dσ)} \le C \Vert ν\Vert, \quad \text{ for all } ν\in \mathcal{M}^+(Ω), \] along with its weak-type analogue, for $0 < q < 1$, where $\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{M}^+(Ω)$ denotes the class of positive Radon measures in $Ω$; $σ, ν\in \mathcal{M}^+(Ω)$, and $||ν||=ν(Ω)$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $σ$ for which such an embedding holds. The strong-type $(1,q)$-ineq…
Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities
University of Connecticut（康涅狄格大学）；Hanoi University of Science and Technology（河内科学技术大学）；Memorial University of Newfoundland（纽芬兰纪念大学）；FPT University（FPT大学）
