Seonghak Kim

CONVEX INTEGRATION AND INFINITELY MANY WEAK SOLUTIONS TO THE PERONA-MALIK EQUATION IN ALL DIMENSIONS ∗

We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baires category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.

On Lipschitz solutions for some forward-backward parabolic equations. II: The case against Fourier

As a sequel to the paper Kim and Yan (Ann Inst H Poincaré Anal Non Linéaire. doi:10.1016/j.anihpc.2017.03.001, 2017), we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward–backward diffusion equations with diffusion fluxes violating Fourier’s inequality.

Strichartz estimates for the magnetic Schrödinger equation with potentials V of critical decay

ABSTRACT We study the Strichartz estimates for the magnetic Schrödinger equation in dimension n≥3. More specifically, for all Schrödinger admissible pairs (r,q), we establish the estimate when the operator H = −ΔA+V satisfies suitable conditions. In the purely electric case A≡0, we extend the class of potentials V to the Fefferman–Phong class. In doing so, we apply a weighted estimate for the Schrödinger equation developed by Ruiz and Vega. Moreover, for the endpoint estimate of the magnetic case in ℝ3, we investigate an equivalence and find sufficient conditions on H and r for which the equivalence holds.

Convex Integration for Scalar Conservation Laws in One Space Dimension

In the absence of the entropy condition, we construct an

Radial weak solutions for the Perona-Malik equation as a differential inclusion

L^\infty

On Lipschitz solutions for some forward-backward parabolic equations

solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits fine-scale oscillations in the i...