Infinity-Laplacian type equations and their associated Dirichlet problems

ABSTRACT

In this paper, we study the Dirichlet problem associated with infinity-Laplacian type equations that may exhibit anisotropic character. We identify a broad class of nonlinearities for which the problem may or may not admit viscosity solutions for any continuous boundary data. We also discuss comparison with Finsler cones, which may be of independent interest.

COMMUNICATED BY:

• D. Repovš

KEYWORDS:

• Infinity-Laplacian type equation
• Finsler–Minkowski norm
• quadratic Finsler cones
• Dirichlet problem

AMS SUBJECT CLASSIFICATIONS:

• 35A01
• 35J60
• 35J70
• 35J75

Acknowledgments

The authors would like to thank the anonymous referees for their comments and for their suggestions that helped correct some misprints in the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The condition $b\ge c^{+}{\delta }_{\mathcal{O}}^{(+)}$ is needed to ensure that ${\mathrm{\Delta }}_{\mathrm{F};\mathrm{\infty }}^{\mathrm{N}}\psi =\kappa -c^{-}$ in $\mathcal{O}\subseteq {\mathrm{\Gamma }}^{+}$. If F satisfies (F-2)${}^{\ast }$, then the restriction on b is unnecessary as, according to Corollary 3.11, the equation ${\mathrm{\Delta }}_{\mathrm{F};\mathrm{\infty }}^{\mathrm{N}}\psi =k-c^{-}$ holds in any open set $\mathcal{O}$ that does not contain z.

2 In a forthcoming paper, it will be shown that the solution is unique when f is identically zero or never vanishes and does not change sign.

3 For instance when $f(x,t,p)={\mathrm{e}}^{t|p|}$ we note that $g\left(t\right)=\left\{\begin{array}{l}0\text{if}t<0\\ 1\text{if}t\ge 0.\end{array}\right.$

Additional information

Funding

This work was supported by ISP (International Science Program) of Uppsala University, Sweden.