Related papers: Regularity and uniqueness results for generated Ja…
Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, $u_t = L(u_x)_x$, where $L$ is merely a monotone function. We also expose the basic properties of solutions,…
In this paper, we prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its…
Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a…
This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn-Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the…
In this note, we give an introduction to the concept of maximal $L^p$-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the…
We study some regularity issues for solutions of non-autonomous obstacle problems with $(p,q)$-growth. Under suitable assumptions, our analysis covers the main models available in the literature.
We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve…
We introduce a weak notion of $2\times 2$-minors of gradients of a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to…
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function…
We give a uniqueness result in dimension 2 for the solutions to an equation on compact Riemannian surface without boundary.
We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$ is a…
We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we…
We present new exact solutions for two-dimensional geometries generated by continuous distributions of topological defects within a conformal metric framework. By reformulating Einstein's equations in two dimensions as a Poisson equation…
We are interested in existence results for second order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal…
We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we…
We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
The two dimensional Jacobian Conjecture says that a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, is invertible. We show that a morphism $f$ having an invertible Jacobian is invertible, in each of the…
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…
We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.