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Michael Bildhauer

Publications and source records attributed to Michael Bildhauer.

At least 19 recordsLinked to original sources
arXiv2026

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.

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arXiv2026

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.

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arXiv2024

Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth

Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein's theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation \[ {\rm div} \Big[Df(\nabla u)\Big] = 0 \, , \] under which solutions have to be affine functions. Here $f$ is a smooth energy density satisfying $D^2 f>0$ together with a natural growth condition for $D^2 f$.

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arXiv2023

A small remark on Bernstein's theorem

We investigate splitting-type variational problems with some linear growth conditions. For balanced solutions of the associated Euler-Lagrange equation we receive a result analogous to Bernstein's theorem on non-parametric minimal surfaces. Without assumptions of this type, Bernstein's theorem cannot be carried over to the splitting case, which follows from an elementary counterexample. We also include some modifications of our main theorem.

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arXiv2023

Splitting-type variational problems with asymmetrical growth conditions

Splitting-type variational problems \[ \int_Ω\sum_{i=1}^n f_i(\partial_i w) dx \to \min \] with superlinear growth conditions are studied by assuming \[ h_i(t) \leq f''_i(t) \leq H_i(t) \] with suitable functions $h_i$, $H_i$: $\mathbb{R} \to \mathbb{R}^+$, $i=1$, $\dots$ , $n$, measuring the growth and ellipticity of the energy density. Here, as the main feature, a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t) \approx H_i(-t)$ for large $|t|$ is not supposed. Assuming quite weak hypotheses as above, we establish higher integrability of $|\nabla u|$ for local minimizers $u\in L^\infty(Ω)$ by using a Caccioppoli-type inequality with some power weights of negative exponent.

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arXiv2022

An extension of a theorem of Bers and Finn on the removability of isolated singularities to the Euler-Lagrange equations related to general linear growth problems

A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general functional of linear growth depending on the modulus of the gradient. We emphasize that Serrin ([1]) in fact proved the removability of singularities on sets of $(n-1)$-dimensional Hausdorff measure zero in an even more general setting. Our main interest is to generalize the comparison principles as outlined, for instance, in Section 10 of [2] without having the particular geometric structure of minimal surfaces. It turns out that generalized catenoids serve as an appropriate tool for proving our results.

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arXiv2022

On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $Ω\subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function ${\rm dist}(\cdot,\partial Ω)$. The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli's inequality involving small weights

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arXiv2021

Liouville-type results in two dimensions for stationary points of functionals with linear growth

We consider variational integrals of linear growth satisfying the condition of $μ$-ellipticity for some exponent $μ>1$ and prove that stationary points $u$: $\mathbb{R}^2 \to \mathbb{R}^N$ with the property \[ \limsup_{|x|\to \infty} \frac{|u(x)|}{|x|} < \infty \] must be affine functions. The latter condition can be dropped in the scalar case together with appropriate assumptions on the energy density providing an extension of Bernstein's theorem.

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arXiv2021

Some geometric properties of nonparametric $μ$-surfaces in $\mathbb{R}^3$

Smooth solutions of the equation \[ \rm{div}\, \Bigg\{ \frac{g'\big(|\nabla u|\big)}{|\nabla u|} \nabla u \Bigg\} = 0 \] are considered generating nonparametric $μ$-surfaces in $\mathbb{R}^3$, whenever $g$ is a function of linear growth satisfying in addition \[ \int_0^\infty s g''(s) d s < \infty \, . \] Particular examples are $μ$-elliptic energy densities $g$ with exponent $μ> 2$ (see [1]) and the minimal surfaces belong to the class of $3$-surfaces. Generalizing the minimal surface case we prove the closedness of a suitable differential form $\hat{N} \wedge d X$. As a corollary we find an asymptotic conformal parametrization generated by this differential form.

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arXiv2020

Splitting-type variational problems with linear growth conditions

Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: $\mathbb{R}^2 \to \mathbb{R}$, \[ f(ξ_1,ξ_2) = f_1\big( ξ_1 \big) + f_2\big( ξ_2 \big) \, , \] with linear growth. As a main result it is shown that, regardless of a corresponding property of $f_2$, the assumption ($t\in \mathbb{R}$) $c_1 (1+|t|)^{-μ_{1}} \le f_1''(t) \le c_2\, ,\quad 1 < μ_1 < 2\, ,$ is sufficient to obtain higher integrability of $\partial_1 u$ for any finite exponent. We also inculde a series of variants of our main theorem. We finally note that similar results in the case $f$: $\mathbb{R}^n \to \mathbb{R}$ hold with the obvious changes in notation.

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arXiv2020

Variational problems of splitting-type with mixed linear-superlinear growth conditions

Variational problems of splitting-type with mixed linear-superlinear growth conditions are considered. In the twodimensional case the minimizing problem is given by \[ J [w] = \int_Ω \Big[f_1\big(\partial_1 w\big) + f_2\big(\partial_2 w\big)\Big] \,dx \to \min \] w.r.t. a suitable class of comparison functions. Here $f_1$ is supposed to be a convex energy density with linear growth, $f_2$ is supposed to be of superlinear growth, for instance to be given by a $N$-function or just bounded from below by a $N$-function. One motivation for this kind of problem located between the well known splitting-type problems of superlinear growth and the splitting-type problems with linear growth (recently considered in [1]) is the link to mathematical problems in plasticity (compare [2]). Here we prove results on the appropriate way of relaxation including approximation procedures, duality, existence and uniqueness of solutions as well as some new higher integrability results.

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arXiv2019

On a class of variational problems with linear growth and radial symmetry

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.

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arXiv2019

Existence results for generalized EED denoising problems

The joint work of the authors with Marcelo Cárdenas and Joachim Weickert \cite{BCFW:2019_1} on edge-enhancing diffusion inpainting problems leads to the analysis of related denoising problems. Here, a surprisingly broad class of diffusion tensors is admissible to obtain the existence of solutions to EED denoising problems.

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arXiv2019

Existence Theory for the EED Inpainting Problem

We establish an existence theory for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoring missing data in an image as steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. We prove existence of a weak solution by applying the Leray-Schauder Fixed point theorem and show that the set of all possible weak solutions is bounded. Moreover, we demonstrate that under certain conditions, the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.

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arXiv2018

On the local boundedness of generalized minimizers of variational problems with linear growth

We prove local boundedness of generalized solutions to a large class of variational problems of linear growth including boundary value problems of minimal surface type and models from image analysis related to the procedure of TV-regularization occurring in connection with the denoising of images, which might even be coupled with an inpainting process. Our main argument relies on a Moser-type iteration procedure.

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arXiv2018

A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter $L$ among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter $L$. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion $G\subsetneq\mathbb{R}^n$. Furthermore, we give a characterization of those (unbounded) convex subsets of $\mathbb{R}^2$ in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to $\mathbb{R}^n$.

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arXiv2018

On the solvability in Sobolev spaces and related regularity results for a variant of the TV-image recovery model: the vector-valued case

We study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class $W^{1,1}$.

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