인문학
사회과학
자연과학
공학
의약학
농수해양학
예술체육학
복합학
DBpia AI
AI 에이전트
AI 아이디어
AI 리더
AI 뷰어
Citeasy
크롬 서지관리 다운로드
지원사업
학술연구/단체지원/교육 등 연구자 활동을 지속하도록 DBpia가 지원하고 있어요.
커뮤니티
연구자들이 자신의 연구와 전문성을 널리 알리고, 새로운 협력의 기회를 만들 수 있는 네트워킹 공간이에요.
초록·키워드
Abstract The Kirchhoff equation was proposed in 1883 by Kirchhoff [ Vorlesungen über Mechanik , Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s , G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msup> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mo>∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> <m:mi>∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> $\frac{{\partial }^{2}u}{\partial {t}^{2}}+\left(a+b{\int }_{{\Omega}}\vert \nabla u{\vert }^{2}\mathrm{d}x\right){\Delta}u=f\left(x,u\right),$ where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mo form="prefix" movablelimits="false">∑</m:mo> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:msubsup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:mfrac> </m:math> ${\Delta}=-\sum \frac{{\partial }^{2}}{\partial {x}_{i}^{2}}$ is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck.
인공지능 문자 인식 모델을 통해 추출된 텍스트로, 일부 오타나 오류가 포함될 수 있으나 지속적으로 개선 중입니다.
오류를 발견하셨다면 해당 부분을 드래그한 후 ' 를 통해 신고해주세요.
오류를 발견하셨다면 해당 부분을 드래그한 후 ' 를 통해 신고해주세요.
최근 본 자료 전체보기
