Dirichlet problems of a quasi-linear elliptic system

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dc.creatorLi, Gongbo
dc.creatorMa, Li
dc.date2003-11-20
dc.date.accessioned2026-07-07T05:03:04Z
dc.date.available2026-07-07T05:03:04Z
dc.descriptionWe discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $Ω$}, & U|_{\partialΩ}&=ϕ. \end{eqnarray*} Here $Ω$ a smooth bounded domain in $R^n$, $f: R^N\to R$ is a smooth function, $U:Ω\to R^N$ is the unknown vector-valued function, $ϕ:\barΩ\to R^N$ is a given vector-valued $C^2$ function, $f'$ is the gradient of the function $f$ with respect to the variable $U$. Such problems arise in population dynamics and Differential Geometry. The difficulty of studying this problem is that this nonlinear elliptic system does not fit the usual growth condition in M.Giaquinta's book [G] and the natural working space $H^1\cap L^{\infty}(Ω)$ for the corresponding Euler-Lagrange functional does not fit the usual minimization or variational argument. We use the direct method on a convex subset of $H^1\cap L^{\infty}(Ω)$ to overcome these difficulties. Under a suitable assumption on the function $f$, we prove that there is at least one solution to this problem. We also give application of our result to the Dirichlet problem of harmonic maps into the standard sphere
dc.description8 pages
dc.identifierhttps://arxiv.org/abs/math/0311343
dc.identifierhttp://arxiv.org/abs/math/0311343
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/69266
dc.subjectAnalysis of PDEs
dc.subject35J50
dc.titleDirichlet problems of a quasi-linear elliptic system
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