Let $\Omega\subset\mathbb R^3$ be a smooth bounded domain and let $g:\partial\Omega\to\mathbb S^2$ be smooth. We prove that any global minimizer of the Oseen--Frank energy subject to the strong anchoring condition $n=g$ is smooth in a full neighborhood of $\partial\Omega$. The result holds for arbitrary positive splay, twist, and bend constants, without any small-anisotropy assumption. It resolves the boundary-regularity part of Lin and Liu's Problem~(c) for the pure Oseen--Frank problem and for its prescribed smooth magnetic-field perturbation.
\noindent Let \(\Om\subset\R^2\) be a smooth bounded simply connected domain. We prove that all positive solutions of \(-\Delta u=\theta ue^{u^2}\), with \(\theta>0\) and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on \(\Om\). Consequently, the Trudinger--Moser functional ha...
We establish a global uniqueness result at the critical Sobolev regularity for the isotropic Calder\'on problem in all dimensions $n\ge 4$. More precisely, for every $n\ge4$ and every bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, any real-valued, uniformly elliptic scalar conductivity $\gamma\in W^{1,n}(\Omega)...
Lu Chen, Yan Jiang, Hong-Yu Liu et al.· 1 citation
Let $\Omega=G/K$ be an irreducible bounded symmetric domain other than the unit disk, and let $S$ be its Shilov boundary. We study the two-parameter family of Shilov--Poisson transforms between homogeneous line bundles over $S$ and $\Omega$. For a transform $F$ of $L^2$ boundary data, we determine what finite Euclidean...
Motivated by a conjecture of Burq and G\'erard, we investigate quantitative uniqueness principles for functions on $\mathbb T^2$ whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control cond...
We prove that a complete noncompact K\"ahler surface with nonnegative Ricci and nonnegative quadratic orthogonal bisectional curvature is contractible, and hence homeomorphic to $\mathbb{R}^4$, if it is simply connected at infinity. Under positive bisectional curvature, this removes the contractibility assumption from...
Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,\alpha}$, for $\alpha \in (0,1)$. Let $p \geq 2$ and $\beta>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shap...
Luca Barbato, Riccardo Molinarolo· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
