Anisotropic overdetermined problem with capillary boundary condition in a half-space

In this paper, we study a Serrin-type anisotropic overdetermined problem with anisotropic capillary boundary in a half-space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Subscript plus Superscript n"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{n}_{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that a domain on which this partially anisotropic overdetermined problem admits a solution if and only if the domain is a truncated Wulff ball intersecting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential double-struck upper R Subscript plus Superscript n"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial \mathbb {R}^{n}_{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant anisotropic capillary angle, which generalizes the isotropic results of Jia-Lu-Xia-Zhang [Calc. Var. Partial Differential Equations 63 (2024), 23 pp.].