arXiv subjects

Alejandro Vélez-Santiago

Publications and source records attributed to Alejandro Vélez-Santiago.

5 recordsLinked to original sources

arXiv2026

A priori estimates for general elliptic and parabolic boundary value problems over irregular domains

We investigate the realization of a myriad of general local and nonlocal inhomogeneous elliptic and parabolic boundary value problems over classes of irregular regions. We present a unified approach in which either local or nonlocal Neumann, Robin, and Wentzell boundary value problems are treated simultaneously. We establish solvability and global regularity results for both the stationary and time-dependent heat equations governed by general differential operators with unbounded measurable coefficients and various boundary conditions at once, first on a general framework, and then by presenting concrete important examples of irregular domains, Wentzell-type boundary conditions, and nonlocal maps. As a consequence, we develop a priori estimates for multiple differential equations under various situations, which are tied to a large number of applications performed over real world regions, such heat transfer, electrical conductivity, stable-like processes (probability theory), diffusion of medical sprays in the bronchial trees, and oceanography (among many others).

math.AP↗

arXiv2025

Generalized quasi-linear fractional Wentzell problems

Given a bounded $(ε,δ)$-domain $Ω\subseteq\mathbb{R\!}^N$ ($N\geq2$) whose boundary $Γ:=\partialΩ$ is a $d$-set for $d\in(N-p,N)$, we investigate a generalized quasi-linear elliptic boundary value problem governed by the regional fractional $p$-Laplacian $(-Δ)^s_{_{p,Ω}}$ in $Ω$, and generalized fractional Wentzell boundary conditions of type $$C'_{p,s}\mathcal{N}^{p'(1-s)}u+β|u|^{q-2} u+Θ^η_qu\,=\,g\indent\indent\indent\textrm{on}\,\,Γ,$$ where $Θ^η_q$ stands as a nonlocal fractional-type $q$-operator on $Γ$ (also refered as a Besov $q$-map), $C'_{p,s}\mathcal{N}^{p'(1-s)}$ denotes the fractional $p$-normal derivative operator in $Γ$, and $p,\,q$ are two growth exponents acting on the interior and boundary, respectively (which are in general unrelated between each other). We first show that this model equation admits a unique weak solution, which is globally bounded in $\overlineΩ$. Furthermore, given two distinct weak solution related to this boundary value problem with different data values, we establish a priori $L^{\infty}$-estimates for the difference of weak solutions with upper bound depending in the differences of the respective interior and boundary data functions. Additionally, a Weak Comparison Principle is derived, and we conclude by establishing a sort of nonlinear Fredholm Alternative related to this generalized elliptic fractional model equation.

math.AP↗

arXiv2025

Diffusion over ramified domains: solvability and fine regularity

We consider a domain $Ω\subseteq\mathbb{R\!}^{\,2}$ with branched fractal boundary $Γ^{\infty}$ and parameter $τ\in[1/2,τ^{\ast}]$ introduced by Achdou and Tchou \cite{ACH08}, for $τ^{\ast}\simeq 0.593465$, which acts as an idealization of the bronchial trees in the lungs systems. For each $τ\in[1/2,τ^{\ast}]$, the corresponding region $Ω$ is a non-Lipschitz domain, which attains its roughest structure at the critical value $τ=τ^{\ast}$ in such way that in this endpoint parameter the region $Ω$ fails to be an extension domain, and its ramified boundary $Γ^{\infty}$ is not post-critically finite. Then, we investigate a model equation related to the diffusion of oxygen through the bronchial trees by considering the realization of a generalized diffusion equation with inhomogeneous mixed-type boundary conditions. Under minimal assumptions, we first show that the stationary version of the above diffusion equation in uniquely solvable, and that the corresponding weak solution in globally Hölder continuous on $\overlineΩ$. Since we are including the critical case $τ=τ^{\ast}$, this is the first time in which global uniform continuity of weak solutions of a Robin-type boundary value problem is attained over a non-extension domain. Furthermore, after two transitioning procedures, we prove the unique solvability of the inhomogeneous time-dependent diffusion equation, and we show that the corresponding weak solution is globally uniformly continuous over $[0,T]\times\overlineΩ$ for each fixed parameter $T>0$.

math.AP↗

arXiv2023

The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

The aim of this paper is to establish the solvability and global regularity theory for a new class of generalized anisotropic heat-type boundary value problems with (pure) dynamical anisotropic Wentzell boundary conditions. We first prove that the Wentzell operator with the above boundary conditions generates a nonlinear order-preserving submarkovian C_0-semigroup \{T_σ(t)\} over \mathbb{X\!}^{\,r(\cdot)}(\overlineΩ):=L^{r(\cdot)}(Ω)\times L^{r(\cdot)}(Γ) for all measurable functions r(\cdot) on \overlineΩ with 1\leq r^-\leq r^+<\infty. Consequently, the corresponding anisotropic dynamical Wentzell problem is well-posed over \mathbb{X\!}^{\,r(\cdot)}(\overlineΩ). Furthermore, we show that the nonlinear C_0-semigroup \{T_σ(t)\} enjoys a Hölder-type ultracontractivity property in the sense that there exist constants C_1,\,C_2,\,κ>0, and γ\in(0,1), such that |\|T_σ(t)\mathbf{u}_0-T_σ(t)\mathbf{v}_0\||_{_{\infty}}\,\leq\, C_1\,e^{C_2t}t^{-κ}|\|\mathbf{u}_0-\mathbf{v}_0\||^γ_{_{r(\cdot),s(\cdot)}} for every \mathbf{u}_0,\,\mathbf{v}_0\in\mathbb{X\!}^{\,r(\cdot),s(\cdot)}(\overlineΩ) and for all t>0.

math.AP↗

arXiv2023

3D Koch-type crystals

We consider the construction of a family $\{K_N\}$ of $3$-dimensional Koch-type surfaces, with a corresponding family of $3$-dimensional Koch-type ``snowflake analogues" $\{\mathcal{C}_N\}$, where $N>1$ are integers with $N \not\equiv 0 \,(\bmod\,\, 3)$. We first establish that the Koch surfaces $K_N$ are $s_N$-sets with respect to the $s_N$-dimensional Hausdorff measure, for $s_N=\log(N^2+2)/\log(N)$ the Hausdorff dimension of each Koch-type surface $K_N$. Using self-similarity, one deduces that the same result holds for each Koch-type crystal $\mathcal{C}_N$. We then develop lower and upper approximation monotonic sequences converging to the $s_N$-dimensional Hausdorff measure on each Koch-type surface $K_N$, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set $\mathcal{C}_N$. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals $\mathcal{C}_N$, for $N>2$.

math.MG↗