arXiv subjects

Franz Gmeineder

Publications and source records attributed to Franz Gmeineder.

At least 19 recordsLinked to original sources

arXiv2026

A Nitsche method for incompressible fluids with general dynamic boundary conditions

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, nonmonotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip-type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a general Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term.

math.NA↗

arXiv2026

Action-Driven Flows for Causal Variational Principles

We introduce action-driven flows for causal variational principles, being a class of non-convex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the non-convexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximate solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.

math-ph↗

arXiv2026

Traces of Sobolev functions and higher integrability

We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or lifting operator.

math.FA↗

arXiv2026

Non-uniformly elliptic variational problems on BV

We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.

math.AP↗

arXiv2025

On the Singular Limit in Hibler's Sea Ice Model

We establish the existence of energy-driven solutions to the momentum balance equation in Hibler's sea ice model. As a main novelty and different from previous results, we deal with the singular limit and therefore cover the true unregularized Hibler stress. To this end, we introduce an energy-based notion of solution that is able to capture plasticity effects of sea ice. This requires certain relaxations of the Hibler energies and, by the different function space set-up, comes with novel challenges. In particular, we establish a bulk approximation result of the boundary terms in the evolutionary relaxed Hibler energies. This is achieved by developing a novel reduction scheme for nonlinear trace expressions which should be of independent interest. Finally, based on our main results, we classify our findings within a broader concept of solutions that is applicable to the non-constant mass case too.

math.AP↗

arXiv2025

Curl Measure Fields, the Generalized Stokes Theorem and Vorticity Fluxes

We introduce and analyze the class $\mathscr{CM}^{p}$ of curl-measure fields that are $p$-integrable vector fields whose distributional curl is a vector-valued finite Radon measure. These spaces provide a unifying framework for problems involving vorticity. A central focus of this paper is the development of Stokes-type theorems in low-regularity regimes, made possible by new trace theorems for curl-measure fields. To this end, we introduce Stokes functionals on so-called good manifolds, defined by the finiteness of manifold-adapted maximal operators. Using novel techniques that may be of independent interest, we establish results that are new even in classical settings, such as Sobolev spaces or their curl-variants $\mathrm{H}^{\mathrm{curl}}(\mathbb{R}^{3})$, which arise, for example, in the study of Maxwell's equations. The sharpness of our theorems is illustrated through several fundamental examples.

math.AP↗

arXiv2025

On Korn inequalities with lower order trace terms

We give an elementary estimate that entails and generalises numerous Korn inequalities scattered in the literature. As special instances, we obtain general Korn-type inequalities involving normal or tangential trace components, or lower dimensional trace integrals.

math.AP↗

arXiv2025

Rate-form equilibrium for an isotropic Cauchy-elastic formulation: Part I: modeling

We derive the rate-form spatial equilibrium system for a nonlinear Cauchy elastic formulation in isotropic finite-strain elasticity. For a given explicit Cauchy stress-strain constitutive equation, we determine those properties that pertain to the appearing fourth-order stiffness tensor. Notably, we show that this stiffness tensor $\mathbb{H}^{\text{ZJ}}(σ)$ acting on the Zaremba-Jaumann stress rate is uniformly positive definite. We suggest a mathematical treatment of the ensuing spatial PDE-system which may ultimately lead to a local existence result, to be presented in part II of this work. As a preparatory step, we show existence and uniqueness of a subproblem based on Korn's first inequality and the positive definiteness of this stiffness tensor. The procedure is not confined to Cauchy elasticity, however in the Cauchy elastic case, most theoretical statements can be made explicit. Our development suggests that looking at the rate-form equations of given Cauchy-elastic models may provide additional insight to the modeling of nonlinear isotropic elasticity. This especially concerns constitutive requirements emanating from the rate-formulation, here being reflected by the positive definiteness of $\mathbb{H}^{\text{ZJ}}(σ)$.

math.AP↗

arXiv2024

Gradient integrability for bounded $\mathrm{BD}$-minimizers

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $\mathrm{BD}(Ω)$ have weak gradients in $\mathrm{L}_{\mathrm{loc}}^{1}(Ω;\mathbb{R}^{n\times n})$. This is achieved for the sharp ellipticity range that is presently known to yield $\mathrm{W}_{\mathrm{loc}}^{1,1}$-regularity in the full gradient case on $\mathrm{BV}(Ω;\mathbb{R}^{n})$. As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $\mathrm{BD}(Ω)$.

math.AP↗

arXiv2024

Extensions of divergence-free fields in $\mathrm{L}^{1}$-based function spaces

We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying $\mathrm{L}^{1}$-boundedness. While previous results as in Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] for $\mathrm{L}^{p}$-based function spaces, $1<p<\infty$, rely on PDE approaches, basic principles from harmonic analysis rule out such strategies in the $\mathrm{L}^{1}$-context. By means of a novel method adapted to the divergence-free constraint via differential forms, we establish the existence of such extension operators in the $\mathrm{L}^{1}$-based situation. This applies both to the case of convex domains, where a global extensions can be achieved, as well as to the Lipschitz case, where a local extension can be achieved. Being applicable to $1<p<\infty$ too, our method provides a unifying approach to the cases $p\in\{1,\infty\}$ and $1<p<\infty$. Specifically, covering the exponents $p\in\{1,\infty\}$, this answers a borderline case left open by Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] in the affirmative. By use of explicit examples, the assumptions on the underlying domains are shown to be almost optimal.

math.AP↗

arXiv2024

The Biot stress -- right stretch relation for the compressible Neo-Hooke-Ciarlet-Geymonat model and Rivlin's cube problem

The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the non-uniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider a remarkable possibility in this nonlinear scenario: Does symmetric loading lead only to symmetric deformations or also to asymmetric deformations? If so, what can we say about monotonicity for these homogeneous solutions, a property which is less restrictive than the energetic stability criteria of homogeneous solutions for Rivlin's cube problem. For the Neo-Hooke type materials we establish what properties the volumetric function $h$ depending on ${\rm det}\, F$ must have to ensure the existence of a unique radial solution (i.e. the cube must continue to remain a cube) for any magnitude of radial stress acting on the cube. The function $h$ proposed by Ciarlet and Geymonat satisfies these conditions. However, discontinuous equilibrium trajectories may occur, characterized by abruptly appearing non-symmetric deformations with increasing load, and a cube can instantaneously become a parallelepiped. Up to the load value for which the bifurcation in the radial solution is realized local monotonicity holds true. However, after exceeding this value, monotonicity no longer occurs on homogeneous deformations which, in turn, preserve the cube shape.

math.AP↗

arXiv2024

Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities

We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*} \lVert P\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le c\big(\lVert\mathscr{A}[P]\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}+\lVert\mathbb{B}P\rVert_{L^{1}(\mathbb{R}^n)}\big) \end{align*} to hold for all $P\in C_{c}^{\infty}(\mathbb{R}^{n};V)$, where $\mathscr{A}$ is a linear map between finite dimensional vector spaces and $\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the $L^{1}$-norm of the differential expression $\mathbb{B}P$ on the right-hand side, such inequalities generalise previously known estimates to the borderline case $p=1$, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.

math.AP↗

arXiv2023

Korn-Maxwell-Sobolev inequalities for general incompatibilities

We establish a family of coercive Korn-type inequalities for generalised incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work (ArXiv 2206.10373), where we focussed on the case $p=1$ and incompatibilities governed by the matrix curl, the case $p>1$ considered in the present paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalisations of recently proved inequalities by the last two authors and Müller (Calc. Var. PDE 60 (2021), 150) in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher order scenarios as well, our approach equally gives the first and sharp inequalities involving Kröner's incompability tensor $\mathrm{inc}$.

math.AP↗

arXiv2023

Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions

We characterise all linear maps $\mathcal{A}\colon\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}$ such that, for $1\leq p<n$, \begin{align*} \|P\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq c\,\Big(\|\mathcal{A}[P]\|_{L^{p^{*}}(\mathbb{R}^{n})}+\|\mathrm{Curl} P\|_{L^{p}(\mathbb{R}^{n})} \Big) \end{align*} holds for all compactly supported $P\in C_{c}^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n\times n})$, where $\mathrm{Curl} P$ displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different constellations between the ellipticities of $\mathcal{A}$, the integrability $p$ and the underlying space dimensions $n$, especially requiring a finer analysis in the two-dimensional situation.

math.AP↗

arXiv2022

$\mathcal{A}$-harmonic approximation and partial regularity, revisited

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,α}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.

math.AP↗

arXiv2022

Boundary ellipticity and limiting $L^1$-estimates on halfspaces

We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality $$ \|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R}^{n},V) $$ holds. This comes as a consequence of sharp trace theorems on $H=\partial H^+$.

math.AP↗

arXiv2022

Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers

We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;Ω]:=\int_ΩF(\nabla u)\,\mathrm{d} x,\qquad u\colonΩ\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity condition on $F$, our results include pointwise $(p,q)$-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise $(p,q)$-growth conditions, our results extend the previously known exponent range from Schmidt's foundational work (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $p=1$. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca & Malý (Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 309-338 (1997)).

math.AP↗

arXiv2022

Natural annihilators and operators of constant rank over $\mathbb{C}$

Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ by an infinite dimensional space. Under the natural condition of constant rank over $\mathbb{C}$, we establish that the equality of nullspaces on the Fourier symbol level already implies the equality of the nullspaces of the differential operators in $\mathscr{D}'$ modulo polynomials of a fixed degree. In particular, this condition allows to speak of natural annihilators within the framework of complexes of differential operators. As an application, we establish a Poincaré-type lemma for differential operators of constant complex rank in two dimensions.

math.AP↗