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P. R. Stinga

Publications and source records attributed to P. R. Stinga.

At least 19 recordsLinked to original sources
arXiv2026

The equivalence of solutions to fractional and logarithmic Helmholtz equations

We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=λ^su$$ for \textit{some} fixed $0 0$ if and only if $u$ is a solution to the same equation for \textit{all} $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\logλ)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-Δ)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace--Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.

math.AP↗
arXiv2025

Boolos' Hardest Logic Puzzle Ever can be solved in no less than three admissible questions: axiomatic framework and rigorous proof

A formal axiomatic mathematical framework for Boolos' Hardest Logic Puzzle Ever is presented and two theorems about its solvability are proved. By strictly following Boolos' instructions (in particular, the requirement that all gods are always obliged to answer), the novel concept of \textit{admissible questions} for the puzzle is introduced. It is then rigorously proved that Boolos' original puzzle can be solved, in an absolute deterministic way, in no less than three yes-no admissible questions. However, this does not mean that one could solve it in less than three admissible questions by just pure \emph{chance}. Hence, such probabilities are computed here as well.

math.HO↗
arXiv2025

Interior Schauder estimates for fractional elliptic equations in nondivergence form

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal regularity assumptions on the coefficients $a^{ij}(x)$. Solutions to the fractional problem are characterized by a local degenerate/singular extension problem. We introduce a novel notion of viscosity solutions for the extension problem and implement Caffarelli's perturbation methodology in the corresponding degenerate/singular Monge--Ampère geometry to prove Schauder estimates in the extension. This in turn implies interior Schauder estimates for solutions to the fractional nonlocal equation. Furthermore, we prove a new Hopf lemma, the interior Harnack inequality and Hölder regularity in the Monge--Ampère geometry for viscosity solutions to the extension problem.

math.AP↗
arXiv2024

A PDE approach to the existence and regularity of surfaces of minimum mean curvature variation

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in $\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is the tangential gradient along $\mathcal{M}$ of the mean curvature $H$ of $\mathcal{M}$ and $dA$ is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results. These are the first analytic results available on the literature for this problem.

math.DG↗
arXiv2024

Sharp extension problem characterizations for higher fractional power operators in Banach spaces

We prove sharp characterizations of higher order fractional powers $(-L)^s$, where $s>0$ is noninteger, ofgenerators $L$ of uniformly bounded $C_0$-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Galé-Miana-Stinga when $0 0$, in terms of boundary derivatives of $U$ at $y=0$, under the sharp hypothesis that $u$ is in the domain of $(-L)^s$. Our results resolve the question of setting up the correct initial conditions that guarantee well-posedness of both extension problems. Furthermore, we discover new explicit subordination formulas for the solution $U$ in terms of the semigroup $\{e^{tL}\}_{t\geq0}$ generated by $L$.

math.AP↗
arXiv2023

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the $C^{0,α}$, $C^{1,α}$ and $C^{2,α}$ regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.

math.AP↗
arXiv2022

Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials and anomalous diffusions

This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials and anomalous diffusions.

math.AP↗
arXiv2022

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

First, we establish the theory of fractional powers of first order differential operators with zero order terms, obtaining PDE properties and analyzing the corresponding fractional Sobolev spaces. In particular, our study shows that Lebesgue and Sobolev spaces with inverse measures (like the inverse Gaussian measure) play a fundamental role in the theory of fractional powers of the first order operators. Second, and motivated in part by such a theory, we lay out the foundations for the development of the harmonic analysis for \emph{inverse} measures. We discover new families of polynomials related to the inverse Gaussian, Laguerre, and Jacobi measures, and characterize them using generating and Rodrigues formulas, and three-term recurrence relations. Moreover, we prove boundedness of several fundamental singular integral operators in these inverse measure settings.

math.CA↗
arXiv2022

Regularity for $C^{1,α}$ interface transmission problems

We study existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with $C^{1,α}$ interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

math.AP↗
arXiv2021

Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality

We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $Ω\subset\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partialΩ$. We show that these fractional operators appear in several applications such as fractional Monge--Ampère equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~Ω\\ u=0&\hbox{on}~\partialΩ\end{cases}$$ is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions $u$ to the fractional problem.

math.AP↗
arXiv2021

Harnack inequalities and Hölder estimates for master equations

We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\timesΩ$$ where $L$ is a divergence form elliptic operator and $Ω\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $Ω$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem.

math.AP↗
arXiv2020

Regularity estimates for nonlocal space-time master equations in bounded domains

We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations $(\partial_t +L)^su = f$ in $\mathbb{R} \times Ω$, where $L$ is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic Hölder spaces in the spirit of Sergio Campanato.

math.AP↗
arXiv2019

Hölder, Sobolev, weak-type and $BMO$ estimates in mixed-norm with weights for parabolic equations

We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,α}_x)$ estimates for $1 0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$$ The coefficients $a(t)=(a^{ij}(t))$ are just bounded, measurable, symmetric and uniformly elliptic. Furthermore, we show strong, weak type and $BMO$-Sobolev estimates with parabolic Muckenhoupt weights. It is quite remarkable that most of our results are new even for the classical heat equation $$\partial_tu-Δu+u=f.$$

math.AP↗
arXiv2019

One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.

math.CA↗
arXiv2018

User's guide to the fractional Laplacian and the method of semigroups

The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and José L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-Δ)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for $(-Δ)^su(x)$. One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on $u$, the validity of the pointwise limits $$\lim_{s\to1^-}(-Δ)^su(x)=-Δu(x)\quad\hbox{and}\quad\lim_{s\to0^+}(-Δ)^su(x)=u(x).$$ The formula for the solution to the Poisson problem $(-Δ)^su=f$ is found through the semigroup approach as the inverse of the fractional Laplacian $u(x)=(-Δ)^{-s}f(x)$ (fundamental solution). We then present the Caffarelli--Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Hölder and Schauder estimates $$(-Δ)^{\pm s}:C^α\to C^{α\mp 2s}$$ are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Hölder and Zygmund spaces with heat semigroups.

math.AP↗
arXiv2018

Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-Δ_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.

math.AP↗