Band 12, Heft 4

In this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below 4⁢π{4\pi}. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x -axis. By working below the energy threshold of 4⁢π{4\pi} and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

We consider an energy functional motivated by the celebrated K13{K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

We consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

We prove the existence of nonnegative weak solutions to a class of second- and fourth-order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space ℝd{{{\mathbb{R}}^{d}}}, for arbitrary d≥1{d\geq 1}. Exploiting a very formal gradient flow structure, the cornerstone of our proof is a modified version of the classical minimizing movement scheme for gradient flows. The mobility function is required to satisfy – at each time point separately – the conditions by which one can define a modified Wasserstein distance on the space of probability densities with finite second moment. The explicit dependency on the time variable is assumed to be at least of Lipschitz regularity. We also sketch possible extensions of our result to the case of bounded spatial domains and more general mobility functions.

We consider the classical non-autonomous variational problem minimize ⁢{F⁢(v)=∫abf⁢(x,v⁢(x),v′⁢(x))⁢dx:v∈Ω},\text{minimize }\biggl{\{}F(v)=\int_{a}^{b}f(x,v(x),v^{\prime}(x))\,\mathrm{d}% x:v\in\Omega\biggr{\}}, where Ω:={v∈W1,1(a,b),v(a)=A,v(b)=B,v(x)∈I}{\Omega:=\{v\in W^{1,1}(a,b),\,v(a)=A,\,v(b)=B,\,v(x)\in I\}}, when the lagrangian f has non-everywhere superlinear growth, in the sense that it can vanish at some x0∈[a,b]{x_{0}\in[a,b]}, or s0∈I{s_{0}\in I}. We prove some sufficient conditions ensuring the coercivity of the functional F . As a consequence, when f is convex with respect to the last variable, the existence of the minimum can be immediately derived.