Let 1≤ n≤ 6, let Omega⊂R^n be a domain, and let f:[0,1)→(0,+∞) be nondecreasing and convex, with lim_t→1^-f(t)=+∞ and int_0^1 f(t),dt=+∞. If u is a stable solution of -Delta u=f(u) in Omega satisfying 0≤ u≤1, must u<1 everywhere in Omega? Here stability means that int_Omega f'(u)varphi^2,dx≤int_Omega|nabla varphi|^2,dx for every test function varphi∈ C_c^1(Omega) for which the expression is defined.

MEMS problem

Let $1\le n\le 6$ , let $\Omega\subset\mathbb{R}^n$ be a domain, and let $f:[0,1)\to(0,+\infty)$ be nondecreasing and convex, with $\lim_{t\to1^-}f(t)=+\infty$ and $\int_0^1 f(t)\,dt=+\infty$ . If $u$ is a stable solution of $-\Delta u=f(u)$ in $\Omega$ satisfying $0\le u\le1$ , must $u<1$ everywhere in $\Omega$ ? Here stability means that $\int_\Omega f'(u)\varphi^2\,dx\le\int_\Omega|\nabla\varphi|^2\,dx$ for every test function $\varphi\in C_c^1(\Omega)$ for which the expression is defined.

References

Primary source

Calculus of Variations and Partial Differential Equations

Additional references

Regularity of stable solutions to the MEMS problem up to the optimal dimension 6 — Calculus of Variations and Partial Differential Equations — Renzo Bruera, Xavier Cabré

elliptic-equations geometric-analysis partial-differential-equations

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Progress summary

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Claimed progress

A new theorem handles six or fewer dimensions under an extra assumption, while the full question for arbitrary nonlinearities and nonradial solutions remains open.

The problem asks whether every stable solution with the stated MEMS-type singularity stays strictly below $1$ when $n\leq 6$ . The retrieved sources provide partial results but no verified resolution under exactly the hypotheses stated.

Known results

In power-law cases $f(u)=(1-u)^{-p}$ , Joseph–Lundgren, Mignot–Puel, Castorina, Esposito–Sciunzi, and Luo–Ye–Zhou obtained partial regularity results and identified singular examples in higher dimensions.

2026 dimension-six theorem and radial claim

Bruera and Cabré prove interior regularity through $n\leq 6$ , and global Dirichlet estimates, assuming a Crandall–Rabinowitz-type condition on $f$ ; the extra condition is unnecessary for $n\leq 2$ . They identify stable singular counterexamples for $n\geq 7$ . A 2026 preprint by Peng claims regularity for stable radial solutions in the unit ball for $2\leq n\leq 6$ without that condition, but this does not cover arbitrary domains or nonradial solutions and remains unverified.

Current status (as of August 2026): The statement is established for $n\leq 6$ only under additional assumptions, while the unrestricted arbitrary-domain problem remains open; the radial claim is unverified.

Sources

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