Interpolating estimates with applications to some quantitative symmetry results

<abstract><p>We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.</p></abstract>

Wolff type potential estimates for stationary Stokes systems with Dini-BMO coefficients

The pointwise gradient estimate for weak solution pairs to the stationary Stokes system with Dini-[Formula: see text] coefficients is established via the Havin–Maz’ya–Wolff type nonlinear potential of the nonhomogeneous term. In addition, we present a pointwise bound for the weak solutions under no extra regularity assumption on the coefficients.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

Elementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Wavelet-based approximations of pointwise bound constraints in Lebesgue and Sobolev spaces

Many problems in optimal control, PDE-constrained optimization and constrained variational problems include pointwise bound constraints on the feasible controls and state variables. Most well-known approaches for treating such pointwise inequality constraints in numerical methods rely on finite element discretizations and interpolation arguments. We propose an alternative means of discretizing pointwise bound constraints using a wavelet-based discretization. The main results show that the discrete, approximating sets converge in the sense of Mosco to the original sets. In situations of higher regularity, convergence rates follow immediately from the underlying wavelet theory. The approach exploits the fact that one can easily transform between a given multiscale wavelet representation and single-scale representation with linear complexity. This allows, for example, a direct treatment of variational problems involving fractional operators, without the need for lifting techniques. We demonstrate this fact with several numerical examples of fractional obstacle problems.

A POINTWISE BOUND ON THE ELECTROMAGNETIC FIELD GENERATED BY A COLLISIONLESS PLASMA

The behaviour of a collisionless plasma is described by the relativistic Vlasov–Maxwell system of equations. A criteria ensuring the existence of smooth solutions was given8by Glassey and Strauss, along with pointwise bounds on the electromagnetic field generated by the particles. In this paper, we obtain an improved bound.