Research Problems in Function Theory — Problem 3.33

Definitions and mathematical statement

Let $\Omega\subset\mathbb R^n$ be bounded and open, $n\ge2$, and $y\in\partial\Omega$. For boundary data $f$, the Perron--Wiener--Brelot method defines upper and lower harmonic envelopes using superharmonic majorants and subharmonic minorants with the required boundary limits. Call $f$ resolutive when these envelopes coincide in a finite harmonic function, denoted $H_f^\Omega$. The point $y$ is $B$-regular for $\Omega$ if, whenever resolutive real boundary data $f$ are bounded in some neighbourhood of $y$ on $\partial\Omega$, their solution $H_f^\Omega$ is bounded in some neighbourhood of $y$ inside $\Omega$. Explicitly, for each such $f$, an $R>0$ with $\sup_{\partial\Omega\cap B(y,R)}|f|<\infty$ must imply the existence of $r>0$ with $\sup_{\Omega\cap B(y,r)}|H_f^\Omega|<\infty$. Call $y$ locally $B$-regular if there is a sequence $r_m\downarrow0$ such that $y$ is $B$-regular for every truncated open set $\Omega\cap B(y,r_m)$, with its own complete boundary and resolutive data. Local $B$-regularity implies $B$-regularity. Is the converse true for every $\Omega$ and $y$? The neighbourhood in the definition may depend on $f$; no uniform bound over all boundary data is required.

Short English statement

Does boundedness regularity for the Dirichlet problem at a boundary point persist on arbitrarily small ball truncations of the domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2303033 (AMR-022-3033), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 3.33. Original source indexed by AMR-022-3033. https://arxiv.org/abs/1809.07200.
• S3 A. Sadi, Some types of regularity for the Dirichlet problem, Section 6. Primary definition and converse question. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/620E7F72E60E8BB1A78F41A02E75F9F5/S0027763000004013a.pdf/some_types_of_regularity_for_the_dirichlet_problem.pdf.