Necessary Conditions for Interpolation by Multivariate Polynomials

Let $$\Omega $$ Ω be a smooth, bounded, convex domain in $${\mathbb {R}}^n$$ R n and let $$\Lambda _k$$ Λ k be a finite subset of $$\Omega $$ Ω . We find necessary geometric conditions for $$\Lambda _k$$ Λ k to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k.

The Keller–Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain $$\varOmega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 . In contrast to previous works it is not assumed that the boundary of $$\varOmega $$ Ω is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

Superlinear Convergence of a Modified Newton's Method for Convex Optimization Problems With Constraints

We consider the constrained optimization problem  defined by: $$f (x^*) = \min_{x \in  X} f(x)\eqno (1)$$ where the function  f : \pmb{\mathbb{R}}^{n} → \pmb{\mathbb{R}} is convex  on a closed bounded convex set X. To solve problem (1), most methods transform this problem into a problem without constraints, either by introducing Lagrange multipliers or a projection method. The purpose of this paper is to give a new method to solve some constrained optimization problems, based on the definition of a descent direction and a step while remaining in the X convex domain. A convergence theorem is proven. The paper ends with some numerical examples.

A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case

If the Radon transform of a compactly supported distribution f ≠ 0 {f\neq 0} in ℝ n {\mathbb{R}^{n}} is supported on the set of tangent planes to the boundary ∂ ⁡ D {\partial D} of a bounded convex domain D, then ∂ ⁡ D {\partial D} must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. 2020, 10.1007/s12220-020-00372-8]. Here we treat the general case.

Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>