arXiv subjects

Eduardo V. Teixeira

Publications and source records attributed to Eduardo V. Teixeira.

At least 19 recordsLinked to original sources

arXiv2026

Qualitative-to-quantitative regularity for degenerate equations

We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,γ}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,β}$ estimates for every $0<β<γ$. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the $L^p$ norm of the solution for every exponent below the sharp Iwaniec--Manfredi threshold. Applied to Xu's recent qualitative $C^{1,1/3}$ regularity for planar infinity-harmonic functions, we obtain uniform interior $C^{1,β}$ estimates for every $0<β<1/3$. For the nonlocal superconductivity equation $|Du|(-Δ)^s u=0$, with $1/2 0$. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.

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arXiv2026

Polynomial growth of complex polynomial Bohnenblust--Hille constants

For complex $m$-homogeneous polynomials, let $D_m$ denote the optimal dimension-free constant in the polynomial Bohnenblust--Hille inequality. We prove that, for every $B>1/2$, there exists $K_B>0$ such that \[ D_m\le K_B m^B,\qquad m\ge1. \] More precisely, we obtain the square-root-scale estimate \[ D_m\le \sqrt m\,\exp\!\bigl(C\sqrt{\log m}\,\log\log m\bigr). \] The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: \[ D_{m,n_m}\longrightarrow 2 \qquad\text{whenever}\qquad \frac{n_m}{m}\longrightarrow1, \] and consequently $\liminf_{m\to\infty}D_m\ge2$. Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.

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arXiv2026

Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm

The Kahane--Salem--Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:(\ell_\infty^n)^m\to\mathbb R$ satisfy \[ \|A\|\le C_m^{\mathrm{sym}}n^{(m+1)/2}. \] For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most $\sqrt{m\log m}\,\sqrt{m!}$; over the real scalar field, an elementary argument gives the sharper order $\sqrt m\,\sqrt{m!}$. We prove \[ C_m^{\mathrm{sym}}\ge \left(\sqrt{\frac2e}+o(1)\right)\frac{\sqrt{m!}}m, \] using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish \[ C_m^{\mathrm{sym}}\le C_0\sqrt{m!}, \] where $C_0$ is absolute, removing the factor $\sqrt m$ from the real upper bound. This estimate follows from a geometric argument in which rigidity for Gram permanents reduces the relevant configurations to sets controlled by Gaussian width. For unrestricted unimodular forms, we also obtain a quantitative rectangular estimate from truncated Hadamard matrices; in equal dimension, the normalized minimum is at most $1+o(1)$ whenever $m=o(n^{5/6})$.

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arXiv2026

Typical Bohnenblust--Hille Ratios

The polynomial Bohnenblust--Hille inequality controls the coefficient $\ell_{2m/(m+1)}$-norm of an $m$-homogeneous polynomial by its supremum norm, with a constant independent of the dimension. We study the associated Bohnenblust--Hille ratio from a probabilistic point of view, by placing normalized surface measure on the Euclidean coefficient sphere. Our results reveal a sharp contrast between the extremal behavior governing the classical Bohnenblust--Hille constants and the typical scale seen in coefficient directions. For arbitrary prescribed monomial supports, the ratio is eventually at most $1$ almost surely, and it tends to zero in spherical measure exactly when the number of monomials tends to infinity. On the full complex polynomial spaces we determine its typical asymptotic scale uniformly in the dimension; in the critical regime $n_m/m\to1$ this gives $2\sqrt2/\sqrt{m\log m}$, in contrast with the nonvanishing extremal scale. For real polynomials, the corresponding ratio tends to zero in spherical measure exactly when the number of variables tends to infinity.

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arXiv2026

Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations

A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly elliptic equations $F(D^2u)=f$, with $f\in L^p$, $p > n$, are locally $C^{1,γ}$ for every $γ< \min\{α_H,σ_p\}$. Here $α_H=α_H(n,λ,Λ)\in(0,1)$ denotes the universal Hölder exponent for gradient regularity of $F(D^2h)=0$, while $σ_p=1-n/p$ is the scaling exponent of the source term. In the source-limited regime $σ_p < α_H$, the singularity of $f$ is the decisive obstruction and the endpoint $γ=σ_p$ is attainable. In the homogeneous-limited regime $α_H\leσ_p$, however, classical theory only yields $γ< α_H$, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When $α_H < σ_p$, we prove that solutions admit pointwise Taylor expansions satisfying $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|\lesssim |x-x_0|^{1+α_H}\left(1+\log\frac{1}{|x-x_0|}\right)^m$. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold $α_H=σ_p$, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields $|u(x)-u(x_0)-Du(x_0)\cdot(x-x_0)|=O\left(|x-x_0|^{1+α_H}\exp\left(A\sqrt{1+\log\frac{1}{|x-x_0|}}\right)\right)$. Both estimates improve the full family of classical sub-endpoint $C^{1,γ}$ bounds, $γ< α_H$, by quantifying differentiability at the limiting homogeneous exponent. The proof introduces a new scale-selection mechanism for endpoint Campanato-type recurrences, suggesting a flexible tool whenever the limiting smoothness is dictated by the homogeneous theory itself. We also discuss the role and possible optimality of the resulting logarithmic defects.

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arXiv2026

Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality

The completely bounded Bohnenblust--Hille inequality of Arunachalam, Dutt, Escudero Gutiérrez and Palazuelos controls coefficient summability with optimal constant one, independently of the ambient dimension. We classify all nonzero equality cases over both the real and complex fields: they are precisely scalar multiples of Hadamard chains supported on equal-sided Cartesian boxes. Thus equality determines both the support and the factorization of the coefficient tensor. For each fixed degree $d\ge2$, we also establish sharp structural stability. A form with completely bounded norm one and deficit $\varepsilon$ lies within $C_d\sqrt\varepsilon$, in the critical coefficient $\ell_{2d/(d+1)}$ norm, of both a flat unimodular path and a normalized unitary chain on the same box. The corresponding unimodular and scaled unitary edges satisfy a matching normalized Frobenius estimate. At zero deficit both approximants coincide with the original Hadamard chain. All constants are independent of the ambient dimension, and the exponent $1/2$ is optimal for both approximations over either field.

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arXiv2026

From qualitative smoothness to uniform estimates for fully nonlinear elliptic equations

We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is that, for compact classes preserved by the natural rescalings of the equation, qualitative and quantitative regularity have the same critical threshold. At second order, mere twice differentiability throughout the corresponding centered hull already forces a uniform $C^{2,α}$-theory for some $α>0$. A one-sided tangent version gives a new partial regularity criterion for a single operator: every point at which a solution is twice differentiable is regular, whereas the singular set has universal positive codimension. The proof combines compactness of the full renormalized class with a scale-adaptive improvement of flatness, converting information available separately at each profile into estimates uniform across the entire class. The argument requires no quantitative control of the assumed smoothness and opens a path in problems where smoothness is visible, but estimates remain out of reach.

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arXiv2026

Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials

We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on $\ell_p^n$. For $m$-homogeneous polynomials, the critical scale is $p=2m$: below this threshold no dimension-free comparison is possible, while at $p=2m$ we obtain $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.

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arXiv2026

Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions

We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $κ=Λ/λ$, then the optimal exponent in the $W^{2,ε}$ regularity theory in the plane is exactly $$ ε_2(κ)=\frac{2}{κ+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\le κ\le 4$ and therefore gives $$ ε_3(κ)=\frac{3}{2κ+1} \qquad(1\leκ\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.

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arXiv2026

The Geometry of Real Anisotropic Bohnenblust--Hille Constants

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust--Hille inequality. For an exponent vector $\mathbf q^{(m)}$, write $C_{\mathbf q^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m d_m/\log m\to\infty$; throughout this regime, their logarithm has the sharp scale $m d_m$. If also $d_m\to0$, then $\log C_{\mathbf q^{(m)}}^{(m)}/(m d_m)$ lies asymptotically in the interval $\left[\frac{\log 2}{4},\frac{2-\log 2-γ}{4}\right]$, where $γ$ is the Euler--Mascheroni constant; this interval has width less than $10^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)/4$ throughout a broad class of anisotropic regimes.

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arXiv2026

Uniform Lipschitz regularity for two-phase singularly perturbed fully nonlinear elliptic equations

We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \fracα{\varepsilon} β\left(\frac{u_\varepsilon}{\varepsilon}\right) \qquad\text{in }B_1\subset\mathbb R^n, $$ where $F$ is uniformly elliptic and $β\in C_c (-1,1)$ is nonnegative. We prove the scale-sharp estimate $$ \|\nabla u_\varepsilon\|_{L^\infty(B_{1/2})} \leq C\left( \|u_\varepsilon\|_{L^\infty(B_1)}+\sqrtα\right), $$ with $C$ depending only on the dimension, the ellipticity constants, and $β$, and independent of $\varepsilon$ and $α$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\varepsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva--Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.

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arXiv2025

On free boundary problems shaped by varying singularities

We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $γ(x)$ and how it changes. Under a mild continuity assumption on $γ(x)$, we prove the optimal regularity of minimizers. Such estimates vary point-by-point, leading to a continuum of free boundary geometries. We also conduct an extensive analysis of the free boundary shaped by the singularities. Utilizing a new monotonicity formula, we show that if the singular power $γ(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,δ}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$.

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arXiv2025

Borderline regularity in singular free boundary problems

In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term $σ$. When $σ$ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if $σ$ is continuous, then minimizers are of class $C^1$ along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.

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arXiv2024

Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels

We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in (0,2)$. Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent $s$. We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems.

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arXiv2024

Oscillatory free boundary problems in stochastic materials

We investigate a class of free boundary problems with oscillatory singularities within stochastic materials. Our main result yields sharp regularity estimates along the free boundary, provided the power of the singularity varies in a Dini-continuous fashion below a certain threshold. We also reveal an interesting repelling estimate preventing the free boundary from touching the region where the singularity power oscillates above the threshold.

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arXiv2024

Optimal domains for elliptic eigenvalue problems with rough coefficients

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e. it is Lipschitz continuous at free boundary points.

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arXiv2024

Regularity in diffusion models with gradient activation

We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient surpasses a given threshold. Our main result concerns the existence of a universal modulus of continuity for $Du$, up to the free boundary. Gradient bounds for the $L^\infty$ norm are proven to be uniform with respect to the degree of degeneracy. Several new ingredients are needed and among the tools introduced in this paper is an improvement of regularity lemma designed to measure the oscillation decay concerning the gradient level-set distance. Applications of the methods are discussed at the end of the paper.

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arXiv2024

Higher regularity of solutions to fully nonlinear elliptic equations

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent regularity limitations dictated by the heterogeneity of the problem. We explore various scenarios, revealing a plethora of improved regularity estimates. Notably, depending on the model's parameters, we establish estimates that transcend the natural regularity regime of the model, from $C^{0,α_0}$ to $C^{1,α_1}$ and further to $C^{2,α_2}$, with the potential for even higher estimates.

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