Interface-Layer Solutions on Perforated Domains

Definitions and mathematical statement

Interface-limit question. Let $\Omega\subset{\mathbb{R}}^N$, $N\ge3$, be a domain with one hole. For sufficiently large $p$, does $-\Delta u=u^p$ admit solutions that converge away from a suitable $(N-1)$-dimensional manifold $M$ to a harmonic function and tend to $1$ on $M$? Since convergence concerns varying exponents, the intended unknown is a family $u_p$, not a single solution at one fixed $p$. In the natural fixed-interface interpretation one seeks a threshold $p_0$, a hypersurface $M$, and functions satisfying\[-\Delta u_p=u_p^p\quad\text{in }\Omega,\qquad u_p\longrightarrow h\quad\text{away from }M,\qquad \Delta h=0\quad\text{in }\Omega\setminus M,\qquad u_p|_M\longrightarrow1\quad(p\to\infty).\] This display records the stated limiting requirements but does not assign an unstated convergence topology, boundary condition, or interface geometry.

The phrase one hole needs a precise topological or boundary description; the source gives neither boundedness nor regularity of $\Omega$. The location, compactness, regularity, possible boundary, and separation properties of $M$ are unspecified. A smooth embedded hypersurface contained in $\Omega$ is one possible meaning, not an added hypothesis. The source also omits whether $u_p$ is positive, nonnegative, or sign-changing, and the allowed exponents. For real nonintegral $p$, the expression $u_p^p$ requires a sign convention or a nonnegative solution; replacing it by $|u_p|^{p-1}u_p$ would change the equation and is not done.

It must further be decided whether a solution is required for every real $p>p_0$ or only along a sequence, whether $M$ is fixed or varies with $p$, whether off-interface convergence is local uniform or in a derivative norm, and whether convergence to one on $M$ is pointwise or uniform. No Dirichlet, Neumann, or other boundary data are supplied, so no definitive boundary-value theorem can be stated from this excerpt.

The supplied triage checked 2026-08-21 labels the question open and reports no located authoritative resolution, but says its extraction omitted both the equation and boundary conditions. The original and curated fields now display the equation; the boundary conditions remain missing. The source layers are therefore not fully reconciled, and their dated open assessment is not independently verified here.

Short English statement

Seek large-exponent solutions with an interface where the limit is one and a harmonic limit away from it. The boundary-value problem, positivity convention, interface geometry, and convergence topology are unspecified. The supplied assessment also describes an earlier extraction lacking the equation now displayed.

Sources

• Local UnsolvedMath registry, permanent ID 30000255, code catalogue entry; Interface-Layer Solutions on Perforated Domains, Oberwolfach Reports Open Problems.
• Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/29, pp. 1601--1678. https://doi.org/10.4171/owr/2005/29. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label original problem report: https://doi.org/10.4171/owr/2005/29.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field concerns restoration of notation, not specification of boundary data. Verify the dated assessment independently.