Boundary behavior of large viscosity solutions to infinity Laplace equations

Zhang, Zhijun

doi:10.1007/s00033-014-0470-1

Cited by 9 publications

(2 citation statements)

References 42 publications

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“…A series of rich and significant information about the boundary behavior of solutions was obtained based on such theory [7][8][9]. In [37], under appropriate structure conditions on the nonlinear term f , Zhang established the following the boundary estimate of large solutions to problem (1-3):…”

Section: Introductionmentioning

confidence: 99%

“…A series of rich and significant information about the boundary behavior of solutions was obtained based on such theory [7–9]. In [37], under appropriate structure conditions on the nonlinear term Zhang established the following the boundary estimate of large solutions to problem (1-3): where is positive and nondecreasing, satisfies and satisfies In [28], the boundary behavior of the boundary blow-up viscosity solutions to problem (1-3) was studied under different conditions on the weight function and the nonlinear term …”

Section: Introductionmentioning

confidence: 99%

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Boundary Blow-Up Solutions to Equations Involving the Infinity Laplacian

LI¹,

LIU

Zhao³

2022

J. Aust. Math. Soc.

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In this paper, we study the boundary blow-up problem related to the infinity Laplacian $$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

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Section: Introductionmentioning

confidence: 99%