김성학 교수

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 but some uncontrollab

We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two imp

이 연구는 농촌마을의 경관계획을 실행하는데 있어 필요한 경관기초단위 도출을 목적으로 진행되었다. 지금까지 다양한 농촌공간의 경관단위 및 경관 요소에 관한 연구들이 진행되어 오고 있지만 실질적인 계획적 측면에서 농촌지역 경관계획을 수립하는데 농촌마을단위의 경관적 정체성을 반영하는데는 그 한계가 있었다. 우리나라 농촌지역의 경관에 관한 계획적 접근은 농업과 주거단위를 통합적으로 접근하는 농촌경관계획과 주거지를 대상으로 하는 마을경관계획으로 크게 구분할 수 있다. 본 연구는 농촌마을 종합개발사업의 한 부분으로 실행되고 있는 경관계획에 마을경관이 가지고 있는 경관 정체성을 반영하고 이를 통하여 마을경관이 가지고 있는 경관적 정체성을 확보하며, 나아가 농촌 경관계획이 보다 현실적 측면에서 실행될 수 있는 방안을 도출하고자 하였다. 연구의 대상지역인 완도군 약산면을 자연적 특성에 기초하여 1:5000 지형도를 이용한 수계단위별 시각적 한계 분석을 실시하고, 분석 결과를 기초로 마을 경관의 가시

We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.

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 study some non-parabolic diffusion problems in one space dimension, where the diffusion flux exhibits forward and backward nature of the Perona–Malik, Höllig or non-Fourier type. Classical weak solutions to such problems are constructed in a way to capture some expected and unexpected properties, including anomalous asymptotic behaviors and energy dissipation or allocation. Specific properties of solutions will depend on the type of the diffusion flux, but the primary method of our study reli

We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases

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 comp

We study the existence and properties of Lipschitz continuous weak solutions to the Neumann boundary value problem for a class of one-dimensional quasilinear forward-backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to be of a forward-backward type that contains two forward-diffusion phases. We prove that, for all smooth initial data, there exists at least one weak solution whose spatial derivative stays in the two forward phases. Also, for