Related papers: Local $L^2$ results for $\bar\partial$: The isolat…

In this paper, we consider the Cauchy-Riemann equation $\bar\partial u= f$ in a new class of convex domains in $\C^n.$ We prove that under $L^p$ data, we can choose a solution in the Lipschitz space $\Lambda_{\alpha},$ where $\alpha$ is an…

In this paper, we extend some results of nonsmooth analysis from Euclidean context to the Riemannian setting. In particular, we discuss the concept and some properties of locally Lipschitz continuous vector fields on Riemannian settings,…

We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lam\'e parameters associated to a linear, isotropic fractional elasticity operator from…

The regularity of the $\bar{\partial}$-problem on the domain $\{|{z_1}|<|{z_2}|<1\}$ in $\mathbb{C}^2$ is studied using $L^2$ methods. Estimates are obtained for the canonical solution in weighted $L^2$-Sobolev spaces with a weight that is…

In this article we consider the classical singular integral operator over a local field with rough kernels. We study the boundedness of such an operator on different function spaces by relaxing the smoothness condition on kernels.

The singularity confinement test is very useful for isolating integrable cases of discrete-time dynamical systems, but it does not provide a sufficient criterion for integrability. Quite recently a new property of the bilinear equations…

We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are:…

Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space…

We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified…

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on…

A generalized variant of the Calder\'on problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown:…

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schr\"odinger-type equations. The approach is based on the observation that generic solutions of…

In this manuscript, we highlight a new phenomenon of complex algebraic singularity formation for solutions of a large class of genuinely nonlinear partial differential equations (PDEs). We start from a unique Cauchy datum, which is…

In characteristic zero, we construct logarithmic resolution of singularities, with simple normal crossings exceptional divisor, using weighted blow-ups.

We consider uniqueness in an inverse Schr\"odinger problem in a bounded domain in $\mathbb{R}^2$ given the Dirichlet-to-Neumann map on part of the boundary. On the remaining boundary we impose a new type of singular boundary condition with…

Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C^N (n less than or equal to N), of codimension one or more in C^N, and endowed with the induced CR structure. We…

We derive sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results…

A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $\Delta u=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $\Delta u=- \kappa(z)…

In this paper, we introduce the notion of oriented faces especially triangles in a connected oriented locally finite graph. This framework then permits to define the Laplace operator on this structure of the 2-simplicial complex. We develop…

We construct modular resolutions of singularities for splitting loci, and use them to show that tame splitting loci have rational singularities. As a corollary of our results and Hurwitz-Brill-Noether theory, we prove that if $C$ is a…