Boundary Conditions Preserving Optimal Hardy–Sobolev Constants

Definitions and mathematical statement

Write $\Omega$ for the domain and $W^{m,p}(\Omega)$ for the Sobolev space appearing in the supplied record. In its usual integer-order interpretation, this is the space of functions whose weak derivatives through order $m$ belong to $L^p(\Omega)$. The record does not specify the admissible values of $m,p,n$, the hypotheses on $\Omega$, or the precise Hardy--Sobolev inequalities under consideration.

For a boundary condition $B$, write formally \[ X_B=\{u\in W^{m,p}(\Omega):u\text{ satisfies }B\}. \] This notation requires an appropriate interpretation of boundary traces, which is not supplied. Let $H_B(p,m,n;\Omega)$ and $S_B(p,m,n;\Omega)$ denote placeholders for the optimal constants in the intended inequalities on $X_B$. These symbols are not definitions of particular variational quotients: the inequalities, weights, exponents, differential operators, normalizations, and optimization conventions are absent from the record.

Source-level classification problem. Determine which boundary conditions $B$, other than the Dirichlet and Navier conditions referred to in the source, preserve the reference optimal constants denoted there by $H(p,m,n)$ and $S(p,m,n)$; formally, the requested preservation is \[ H_B(p,m,n;\Omega)=H(p,m,n),\qquad S_B(p,m,n;\Omega)=S(p,m,n). \] These equalities express the supplied target only after the intended inequalities and reference constants have been specified. The record does not define the higher-order Dirichlet or Navier conditions, establish whether their reference constants coincide in every intended setting, or specify whether preservation is required simultaneously for both constants. It also leaves unspecified the class of competing boundary conditions and whether the assertion is to hold for a fixed domain and parameter tuple or uniformly over an admissible family. Thus a fully quantified mathematical classification problem cannot be recovered from the supplied record alone.

Short English statement

The problem asks which boundary conditions besides Dirichlet and Navier preserve the optimal constants in the intended Hardy--Sobolev embeddings. The supplied record omits the defining inequalities, parameter ranges, domain hypotheses, and boundary trace conventions, so the displayed preservation target remains conditional on those specifications.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001414$, title: Boundary Conditions Preserving Optimal Hardy--Sobolev Constants; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2010; no individual proposer is named.
• Supplied report citation: Optimal Constants in the Theory of Sobolev Spaces and PDEs (2010), Oberwolfach Reports, pp. 325--371, DOI 10.4171/owr/2010/08. Source URL: https://doi.org/10.4171/owr/2010/08. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 8/2010, https://doi.org/10.4171/owr/2010/08.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, attributes the classification question to the report, and states that no definitive general classification was found. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is self-contained and introduces no unsupported source-dependent content. That supplied review label does not resolve the missing mathematical specifications identified here. The report itself has not been independently consulted for this draft.