We study a boundary value problem for the Helmholtz equation with an impedance boundary condition, in two and three dimensions, modelling the scattering of time harmonic acoustic waves by an unbounded rough surface. Via analysis of an equivalent variational formulation we prove this problem to be well-posed when: i) the boundary has the strong local Lipschitz property and the frequency is small; ii) the rough surface is the graph of a bounded Lipschitz function (with arbitrary frequency). An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms, are explicit in terms of the wavenumber k, the geometry of the scatterer and the parameters describing the surface impedance.

We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations. As a consequence, the gradient Ricci solitons that arise from the reduced system are invariant under the action of either an $(n-1)$-dimensional translation group or the pseudo-orthogonal group acting on the corresponding $n$-dimensional pseudo-Euclidean space.

Let R be a compact surface and let Gamma be a Jordan curve which separates R into two connected components Sigma(1) and Sigma(2). A harmonic function h(1) on Sigma(1) of bounded Dirichlet norm has ...

We show that every non-compact weighted composition operator $f \mapsto u\cdot (f\circ\phi)$ acting on a Hardy space $H^p$ for $1 \leq p < \infty$ fixes an isomorphic copy of the sequence space $\ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a copy of the Hilbert space $\ell^2$. These results extend earlier ones obtained for unweighted composition operators.

In the current paper, we study how the speed of convergence of a sequence of\nangles decreasing to zero influences the possibility of constructing a rare\ndifferentiation basis of rectangles in the plane, one side of which makes with\nthe horizontal axis an angle belonging to the given sequence, that\ndifferentiates precisely a fixed Orlicz space.\n

It was shown in Bj\"orn--Bj\"orn--Korte ("Minima of quasisuperminimizers", Nonlinear Anal. 155 (2017), 264-284) that $u:=\min\{u_1,u_2\}$ is a $\overline{Q}$-quasisuperminimizer if $u_1$ and $u_2$ are $Q$-quasisuperminimizers and $\overline{Q}=2Q^2/(Q+1)$. Moreover, one-dimensional examples therein show that $\overline{Q}$ is close to optimal. In this paper we give similar examples in higher dimensions. The case when $u_1$ and $u_2$ have different quasisuperminimizing constants is considered as well.

In this paper, we prove the existence and multiplicity results of solutions with prescribed L 2 -norm for a class of nonlinear Chern-Simons-Schrdinger equations in R 2

We provide a Reifenberg type characterization for m-dimensional C 1 -submanifolds of R n . This characterization is also equivalent to Reifenberg-flatness with vanishing constant combined with suitably converging approximating m-planes. Moreover, a sufficient condition can be given by the finiteness of the integral of the quotient of (r)-numbers and the scale r, and examples are presented to show that this last condition is not necessary.

In this note we study generalized differentiability of functions on a class of fractals in Euclidean spaces. Such sets are not necessarily self-similar, but satisfy a weaker "scale-similar" property; in particular, they include the non self similar carpets introduced by Mackay-Tyson-Wildrick Specifically we identify certain geometric criteria for these fractals and, in the case that they have zero Lebesgue measure, we show that such fractals cannot support nonzero derivations in the sense of Weaver As a result (Theorem 26) such fractals cannot support Alberti representations and in particular, they cannot be Lipschitz differentiability spaces in the sense of Cheeger [3] and Keith

We prove that any metric space X homeomorphic to R 2 with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let Q X be a topological quadrilateral with boundary edges (in cyclic order) denoted by