A maximum principle for mean-curvature type elliptic inequalities

James Serrin

doi:10.4171/jems/59

JournalsjemsVol. 8, No. 2pp. 389–398

A maximum principle for mean-curvature type elliptic inequalities

James Serrin
University of Minnesota, Minneapolis, United States

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Abstract

Consider the divergence structure elliptic inequality

div {A (x, u, D u)} + B (x, u, D u) \geq 0 (1)

in a bounded domain $Ω \subset R^{n}$ . Here

A (x, z, ξ) : K \to R^{n}, B (x, z, ξ) : K \to R, K = Ω \times R^{+} \times R^{n},

and $A$ , $B$ satisfy the following conditions

⟨ A (ξ), ξ ⟩ \geq ∣ ξ ∣ - c (x) z - a (x), ∣ A (x, z, ξ) ∣ \leq Const., B (x, z, ξ) \leq b (x),

for all $(x, z, ξ)) \in K$ , where $a (x)$ , $b (x), c (x)$ are given non-negative functions. Our interest is in the validity of the maximum principle for solutions of (1), that is, the statement that any solution which satisfies $u \leq 0$ on $\partial Ω$ must be a priori bounded above in $Ω$ .

This question arises, in particular, when one is interested in the mean curvature equation

div \frac{D u}{1 + ∣ D u ∣ ^{2}} = n H (x) .

Cite this article

James Serrin, A maximum principle for mean-curvature type elliptic inequalities. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 389–398

DOI 10.4171/JEMS/59