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M. Fuchs

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Preprint Aug 2026

Bernstein's theorem for variational integrals of linear growth and radial structure

We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_{\Omega} g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(t)\,dt<\infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.

M. Fuchs, M. Bildhauer · 0 citations

Preprint Aug 2026

Uniqueness and boundary behaviour of solutions to variational problems with linear growth

We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0)>\frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.

M. Bildhauer, M. Fuchs · 0 citations