Related papers: Minimum principles and a priori estimates for some…
Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems…
We study new examples of translating solitons of the mean curvature flow, especially in Minkowski space. We consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This…
The soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations. Roughly speaking it asserts that a solution with generic initial condition converges to a finite number of…
In this paper, we consider the problem of finding the hypersurface M^n in the Euclidean (n+1)-space R^{n+1} that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically…
We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable…
We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local $C^{1,1}$ estimates…
We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most…
We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our…
We consider localized soliton-like solutions in the presence of a stable scalar condensate background. By the analogy with classical mechanics, it can be shown that there may exist solutions of the nonlinear equations of motion that…
Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of…
We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of…
We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are…
We deal with solitons of the mean curvature flow. The definition of \textit{translating solitons on a light-like direction} in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, \textit{translation surfaces},…
Minimal cubature rules of degree $4n-1$ for the weight functions $$ W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are constructed explicitly and are shown to be closed related to…
In this paper, we consider a translating soliton for the inverse mean curvature flow given as a graph of a function on a domain in a unit sphere whose level sets give isoparametric foliation. First, we show that such function is given as a…
Obstructions to the existence of spacelike solitons depending on the growth of the mean curvature $H$ are proved for Lorentzian products $(M\times \mathbb{R}, \bar g=g_M-dt^2)$ with lowerly bounded curvature. The role of these bounds for…
Canonical work handling distribution shifts typically necessitates an entire target distribution that lands inside the training distribution. However, practical scenarios often involve only a handful of target samples, potentially lying…
All simple translation-invariant valuations on polytopes are classified. As a direct consequence the well-known conditions for translative-equidecomposability are recovered. Furthermore, a simplified proof of the classification of…
The problem of learning a minimal consistent model from a set of labeled sequences of symbols is addressed from a satisfiability modulo theories perspective. We present two encodings for deterministic finite automata and extend one of these…
For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error…