Optimal sets for a class of minimization problems with convex constraints

Solution by convex minimizationof the Cauchy problem forhyperbolic systems of conservation laws with convex entropy

We show that, for first-order systems of conservation laws with a strictly convex entropy,in particular for the very simple so-called "inviscid" Burgers equation,it is possible to address the Cauchy problem by a suitable convex minimizationproblem, quite similar to some problems arising in optimal transport or variational mean-field game theory.In the general case, we show that smooth, shock-free, solutions can be recoveredon some sufficiently small interval of time. In the special situation of the Burgers equation, we furthershow that every "entropy solution" (in the sense of Kruzhkov)includi…

Some minimization problems in the class of convex functions with prescribed determinant

We consider minimizers of linear functionals of the type $$L(u)=\int_{\p Ω} u \, d σ- \int_Ω u \, dx$$ in the class of convex functions $u$ with prescribed determinant $\det D^2 u =f$. We obtain compactness properties for such minimizers and discuss their regularity in two dimensions.

Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization

We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope.

Existence of variational solutions to nonlocal evolution equations via convex minimization

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0. \end{align*} The approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Un…

可微凸函数的单调逼近及其在一般极小化问题中的应用

Monotone approximation of differentiable convex functions with applications to general minimization problems

本文在极小增长与强制假设下研究非自治能量的极小元，证明其满足相关 Euler--Lagrange 方程或不等式，核心工具是凸 $C^1$-能量的延拓结果。

Stochastic PDEs via convex minimization

We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle [50] to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approch via convex optimization to the approximation of nonlinear …