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- W2611111582 abstract "We consider the problem of finding and describing minimisers of the Rayleigh quotient $$begin{aligned} Lambda _infty , :=, inf _{uin mathcal {W}^{2,infty }(Omega )setminus {0} }frac{Vert Delta uVert _{L^infty (Omega )}}{Vert uVert _{L^infty (Omega )}}, end{aligned}$$ where $$Omega subseteq mathbb {R}^n$$ is a bounded $$C^{1,1}$$ domain and $$mathcal {W}^{2,infty }(Omega )$$ is a class of weakly twice differentiable functions satisfying either $$u=0$$ on $$partial Omega $$ or $$u=|mathrm {D}u|=0$$ on $$partial Omega $$ . Our first main result, obtained through approximation by $$L^p$$ -problems as $$prightarrow infty $$ , is the existence of a minimiser $$u_infty in mathcal {W}^{2,infty }(Omega )$$ satisfying $$begin{aligned} left{ begin{array}{ll} Delta u_infty , in , Lambda _infty mathrm {Sgn}(f_infty ) &{} text { a.e. in }Omega , Delta f_infty , =, mu _infty &{} text { in }mathcal {D}'(Omega ), end{array} right. end{aligned}$$ for some $$f_infty in L^1(Omega )cap BV_{text {loc}}(Omega )$$ and a measure $$mu _infty in mathcal {M}(Omega )$$ , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue $$Lambda _infty $$ on the domain, establishing the validity of a Faber–Krahn type inequality: among all $$C^{1,1}$$ domains with fixed measure, the ball is a strict minimiser of $$Omega mapsto Lambda _infty (Omega )$$ . This result is shown to hold true for either choice of boundary conditions and in every dimension." @default.
- W2611111582 created "2017-05-12" @default.
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- W2611111582 date "2017-11-20" @default.
- W2611111582 modified "2023-10-16" @default.
- W2611111582 title "The eigenvalue problem for the $$infty $$-Bilaplacian" @default.
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- W2611111582 doi "https://doi.org/10.1007/s00030-017-0492-4" @default.
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