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Eduardo V. Teixeira

Publications and source records attributed to Eduardo V. Teixeira.

At least 55 records · Page 3Linked to original sources
arXiv2015

Geometric approach to nonvariational singular elliptic equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(γ-1)$, for $0 < γ< 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.

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arXiv2015

Geometric regularity estimates for elliptic equations

Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems, mathematical physics, economics, etc. Within the general field of Partial Differential Equations, regularity theory places itself in the very core, by bridging the notion of weak solutions (often found by energy methods or probabilistic interpretations) to the classical concept of solutions. In this article we discuss about a new systematic approach, termed geometric tangential analysis, for addressing regularity estimates for elliptic and parabolic problems.

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arXiv2015

Sharp hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach

We establish sharp $W^{2,p}$ regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator $F$. By means of geometric tangential methods, we show that if the {\it recession} of the operator $F$ -- formally given by $F^*(M):=\infty^{-1} F(\infty M)$ -- is convex, then any viscosity solution to the original equation $F(D^2u) = f(x)$ is locally of class $W^{2,p}$, provided $f\in L^p$, $p>d$, with appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they are new even under convexity of the frozen coefficient operator, $M\mapsto F(x_0, M)$, as oscillation is measured only at the recession level. The methods further yield BMO regularity of the hessian, provided the source lies in that space. As a final application, we establish the density of $W^{2,p}$ solutions within the class of all continuous viscosity solutions, for generic fully nonlinear operators $F$. This result gives an alternative tool for treating common issues often faced in the theory of viscosity solutions.

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arXiv2015

Free transmission problems

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, $\mbox{L}_{+}$ and $\mbox{L}_{-}$, within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and $L^{\infty}$ bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.

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arXiv2015

Global Monge-Ampere equation with asymptotically periodic data

Let $u$ be a convex solution to $\det(D^2u)=f$ in $\mathbb R^n$ where $f\in C^{1,α}(\mathbb R^n)$ is asymptotically close to a periodic function $f_p$. We prove that the difference between $u$ and a parabola is asymptotically close to a periodic function at infinity, for dimension $n\ge 3$.

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arXiv2013

Optimal gradient continuity for degenerate elliptic equations

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nabla u(X) = 0 \}$. The innovative feature of our main result concerns its optimality -- the sharp, encoded smoothness aftereffects of the operator. Such a quantitative information usually plays a decisive role in the analysis of a number of analytic and geometric problems. Our result is new even for the classical equation $|\nabla u | \cdot Δu = 1$. We further apply these new estimates in the study of some well known problems in the theory of elliptic PDEs.

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arXiv2013

Hessian continuity at degenerate points in nonvariational elliptic problems

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such context, it has been a common accepted aphorism that the continuity of the Hessian of a solution could never be superior than the continuity of the medium. Notwithstanding, in this article we show that solutions to uniformly elliptic, linear equations with $C^{0,\barε}$ coefficients are of class $C^{2,α}$, for any $0 < \barε \ll α< 1$, at Hessian degenerate points, $\mathscr{H}(u):=\{X \suchthat D^2u(X) = 0\}$. In fact we develop a more general regularity result at such Hessian degenerate points, featuring into the theory of fully nonlinear equations. Insofar as the optimal modulus of continuity for the Hessian is concerned, the result of this paper is the first one in the literature to surpass the inborn obstruction from the sharp Schauder {\it a priori} regularity theory.

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arXiv2013

A geometric tangential approach to sharp regularity for degenerate evolution equations

That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous $p-$Laplace equation $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ are $C^{0,α}$, for some $α\in (0,1)$, is known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Hölder exponent $α$ in terms of $p, q, r$ and the space dimension $n$. We show in this paper that $$ α= \frac{(pq-n)r-pq}{q[(p-1)r-(p-2)]}, $$ using a method based on the notion of geometric tangential equations and the intrinsic scaling of the $p-$parabolic operator. The proofs are flexible enough to be of use in a number of other nonlinear evolution problems.

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arXiv2013

Regularity for degenerate two-phase free boundary problems

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_γ\to $ min, ruled by nonlinear, $p$-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to $\mathcal{J}_γ$ becomes singular along the free interface $\{u= 0\}$. The degree of singularity is, in turn, dimed by the parameter $γ\in [0,1]$. For $0< γ< 1$ we show local minima is locally of class $C^{1,α}$ for a sharp $α$ that depends on dimension, $p$ and $γ$. For $γ= 0$ we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.

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arXiv2013

Regularity for quasilinear equations on degenerate singular sets

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

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arXiv2012

Sharp regularity for general Poisson equations with borderline sources

This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + ε}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is based on powerful Caffarelli-type compactness methods and it can be employed in a number order situations.

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arXiv2011

Universal moduli of continuity for solutions to fully nonlinear elliptic equations

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,α}$ regularity estimates are delivered when $f\in L^{n+ε}$. The limiting upper borderline case, $f\in L^\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \in C^{1,\mathrm{Log-Lip}}$, provided $f$ has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior \textit{a priori} estimates on the $C^{0,\frac{n-2ε}{n-ε}}$ norm of $u$ based on the $L^{n-ε}$ norm of $f$, where $ε$ is the Escauriaza universal constant. The exponent $\frac{n-2ε}{n-ε}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-ε$, we also obtain the exact, improved sharp Hölder exponent of continuity.

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arXiv2011

Regularity and geometric estimates for minima of discontinuous functionals

In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim χ_{\{u > 0 \}}$. The Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate and nondegeneracy of minima. We also address weak and strong regularity properties of free boundary. We show the set $\{u > 0 \}$ has locally finite perimeter and that the reduced free boundary, $\partial_\text{red} \{u > 0 \}$, has $\mathcal{H}^{n-1}$-total measure. For more specific problems that arise in jet flows, we show the reduced free boundary is locally the graph of a $C^{1,γ}$ function.

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arXiv2009

Asymptotic behavior of solutions to the $σ_k$-Yamabe equation near isolated singularities

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and $σ_k$ curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

math.AP↗
arXiv2009

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1<p and 1<=q, a linear operator T from l_p to l_q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p=1). For 1<p (and q different from p), as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T from l_p to l_q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p=q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.

math.FA↗
arXiv2008

Optimal design problems in rough inhomogeneous media. Existence theory

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove the existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to enjoy the appropriate weak geometry and we establish the existence of an optimal design for general convex optimal design problems with volume constraints for all dimensions.

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