Sharp Geometric Constants for Simply Connected Planar Domains

Definitions and mathematical statement

For reference, let $D\subsetneq{\mathbb{R}}^2\cong{\mathbb{C}}$ be a nonempty simply connected open set. Define its distance to the boundary and its inradius by \[ \delta_D(x)=\inf_{y\in\partial D}|x-y|, \qquad r(D)=\sup_{x\in D}\delta_D(x)\in(0,\infty]. \] The following conventional quantities describe the objects named in the supplied record, without resolving its missing conventions. For a real-valued smooth function $u$ with compact support in $D$, the usual Hardy weighted integral and Dirichlet energy are \[ H_D(u)=\int_D\frac{|u(x)|^2}{\delta_D(x)^2}\,{\,\mathrm{d}} x, \qquad \mathcal E_D(u)=\int_D|\nabla u(x)|^2\,{\,\mathrm{d}} x. \] With the operator convention $-\Delta$, define the bottom of the Dirichlet spectrum by \[ \lambda_D=\inf_{0\ne u\in C_c^\infty(D;{\mathbb{R}})} \frac{\mathcal E_D(u)}{\int_D|u(x)|^2\,{\,\mathrm{d}} x}. \] For an unbounded domain, this infimum need not be an eigenvalue; thus the source's phrase "first Dirichlet eigenvalue" requires clarification for its stated domain class.

For planar Brownian motion $(B_t)_{t\ge0}$ started at $x\in D$, write \[ \tau_D=\inf\{t\ge0:B_t\notin D\}, \qquad T_D(x)=\mathbb E_x[\tau_D]\in[0,\infty]. \] The record does not specify the Brownian generator, so no time normalization is selected here. Write $\rho_D(z)|{\,\mathrm{d}} z|$ for a hyperbolic metric on $D$; its curvature normalization is likewise unspecified.

Requested problem, with unresolved qualifications. Identify the optimal universal constants $C_1,C_2,C_3,C_4$, and describe the geometry of domains attaining or approaching the corresponding sharp bounds, in four inequalities for arbitrary simply connected planar domains. In the supplied order these concern: a Hardy inequality involving distance to the boundary; the first Dirichlet eigenvalue and the inradius; expected Brownian exit time and the inradius; and hyperbolic metric density and the inradius.

Formulation gap. The supplied record does not display any of the four inequalities. Consequently it does not specify their directions, the placement of the constants, the function quantifier in the Hardy inequality, or whether exit time and metric density are evaluated at a point or aggregated over the domain. The definitions above are reference conventions, not a reconstruction of those missing inequalities. The record also does not say whether domains must be proper, bounded, or have finite inradius. The properness assumption used for these definitions is needed for the stated hyperbolic object and is not an established restriction of the intended problem: ${\mathbb{C}}$ is simply connected but has no hyperbolic metric of constant negative curvature. A unique fully quantified sharp-constant problem therefore cannot be recovered from the supplied text alone.

Short English statement

The supplied problem asks for four best constants and their extremal domain geometries for simply connected planar domains. The four inequalities and several essential conventions are missing, so this draft records the named quantities and the unresolved formulation issues. The registry labels the problem open; that status is supplied curation, not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $30001161$, title "Sharp Geometric Constants for Simply Connected Planar Domains," source problem code catalogue entry. The record gives proposed year 2009 and does not name a proposer.
• Supplied report citation: Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. https://doi.org/10.4171/owr/2009/06. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature source, labeled "Bañuelos, four unknown constants": https://hopf.math.purdue.edu/~banuelos/Papers/4-constants.pdf. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the convex-domain constants are known, while the sharp constants and extremal geometries for the full simply connected class are not all identified and the dedicated survey continues to list unknown cases. Both links above are listed as sources checked by that curation. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and reports that the mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion; no source visit or independent verification is claimed in this draft.