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Let $K\subset\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \( \overline M=\R^m\setminus\operatorname{int}K \) be the allowed target. We prove that if $\partial K$ is of class $C^{1,1}$, then every $\overline M$-valued local minimizer of the Dirichlet energy $E(u;B):...
We classify isolated singularities of classical solutions to the capillary equation with negative gravity, \[ \operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u \qquad\text{in }B_R\setminus\{0\}\subset\mathbb R^n, \qquad n\ge2, \] without a priori assumptions on symmetry, sign, one-sided boundedness, or blow-up rate. Ever...
We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(\Omega)\) be a weak \(p\)-harmonic function in a domain \(\Omega\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locall...
Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}....
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