E. Teixeira s Research Work

Transcription

1 Work E. Teixeira s Research EDUARDO V. TEIXEIRA Professor of Mathematics September 2015 I. OVERVIEW My research activities are grounded in the theory of partial differential equations (PDE) and their applications, with particular attention to problems involving diffusive processes. Such models appear naturally in the mathematical formulation of a number of problems in pure and applied sciences. Understanding their intrinsic regularity theories is of central importance in the analysis of problems in physics, biology, fluid dynamics, economics and financial mathematics, probability, differential geometry, etc. When diffusive processes involve discontinuous changes and diverse laws in a priori unknown regions, new mathematical challenges arise. This refers to the theory of free boundary problems and the mathematical analysis involved in addressing them has promoted significant advances in the PDE theory with a wide range of application. While my research has been greatly influenced by my academic mentors, in particular my doctoral adviser Dr. Luis Caffarelli, over the past decade it has acquired a signature of its own. By means of new ideas and methods it has fostered an enhanced understanding on topics which include the theory of fully nonlinear elliptic equations, nonvariational free boundary problems, nonlinear potential theory, singular and degenerate PDE, among others. A feature worth noting is the fluency with which my research imports and adapts free boundary tools and insights to different contexts. By way of example, I have recently addressed classical questions from nonlinear potential theory, using this approach which led to a plethora of unanticipated results. Subsequently I shall provide a summary of select contributions I have made to two main research fields, viz., free boundary problems and regularity theory for diffusive models. As just noted, an admixture of ideas, methods, and tools encountered in the following sections is not coincidental. I will then close with a glance at some projects I am committed to carry out within the next few years. II. CONTRIBUTIONS TO THE THEORY OF FREE BOUNDARY PROBLEMS I. Optimal design My first notable scientific contribution to the field of free boundary problems concerns a classic question from thermodynamics: given an n-dimensional body D and a prescribed volume-budget ι > 0, which is the most efficient way to isolate D?". The mathematical system that models this problem is flexible and applies to a broad spectrum of problems in mathematical physics, biology, finance, and other fields. Accordingly, the problem has attracted the attention of multi-disciplinary researchers for some time. However, it was only in the 1980s, following developments in geometric measure theory by Alt and Caffarelli [1] that the problem could be properly addressed since such optimization problems involve minimization of integral flows defined over measurable sets with volumetric constraints. Using controlled penalty combined with tools from harmonic analysis, in [Calc. Var. Partial Differential Equations 24 (2005)], I prove the existence and regularity of solutions and their free boundaries. A pioneering feature of this work relies on the fact that the flux equation proven to hold along the free boundary has a nonlocal character. Qualitative and geometric properties of the free boundary 1

2 were subsequently delineated in [Interfaces Free Bound. 9 (2007)]. The optimal insulation problem in heterogeneous and discontinuous media is relevant to diverse applications and presents difficult mathematical challenges. The existence and regularity of sharp configurations for this general class of problems was studied in [Amer. J. Math. 132 (2010)]. In collaboration with J. Rossi, [Trans. Amer. Math. Soc. 364 (2012)], we analyzed the convergence and stability of optimal design problems when the degeneracy parameter tends to infinity. The limiting problem is shown to be governed by the infinity laplacian operator. Recently, during the postdoctorate studies of R. Teymurazyan, we prove the existence and regularity estimates for the fractional optimal design problem, where the energy is given by a fractional Sobolev norm; a study which will be published in J. London Math. Soc. II. Singularly perturbed problems In parallel to the investigation of optimal design problems, I have also examined singularly perturbed partial differential equations for high-energy activation models, such as in flame-propagation theory. The study of such regularized free boundary problems has yielded significant scientific breakthroughs since the pioneering work of Berestycki, Caffarelli, and Nirenberg in the early 1990s. Partnering with Moreira, [Calc. Var. Partial Differential Equations 29 ( 2007)], we analyzed heterogeneous singularly perturbed equations related to super-determined boundary problems. Because the leading operator is in divergence form, we could treat the problem by variational means. In [Ann. Inst. H. Poincare Anal. Non Linéaire 25 (2008)], I studied this class of problems for equations involving nonzero drift terms. The principal difficulty arises from the fact the Euler-Lagrange functional cannot be associated with the problem in principle. This obstacle was overcome by proving that one can select approximating solutions u ɛ with a kind of minimization property for a new functional derived from multivalued fixed-point theorems. When the governing operator is completely nonlinear and nonvariational, the problem becomes significantly more complex. The fully-nonlinear singular perturbation problem was the theme of the doctoral thesis of my first Ph.D. student, G. Ricarte. In our joint work, [J. Funct. Anal. 261 (2011)], we prove, among several other analytic and geometric properties, that the free boundary condition is governed by a new operator, F, obtained via nonstandard blow-up arguments on the family of elliptic equations generated by the original operator F. Subsequently, jointly with Ricarte and Araujo, we analyzed nonvariational singularly perturbed equations involving two free boundaries, viz., what we term the physical transition" {u ɛ ɛ} and nonphysical free boundary" { u ɛ = 0}. The main difficulty was to show that these boundary sets do not intersect in measure. III. Obstacle problems A classic question from mathematical physics refers to the equilibrium position of an elastic membrane reposing on an obstacle, often called the obstacle problem. The mathematical model used for this class of problem is flexible and applicable in many contexts. In a joint study with A. Swiech [Adv. Math. 220 (2009)], we analyze obstacle problems in infinite-dimensional environments, establishing existence, uniqueness, and sharp regularity estimates for the solution. In a study conducted with Rossi and Urbano, to be published in Interfaces Free Bound., we provide a detailed analysis of obstacle problems governed by the infinity laplacian operator. The main result proved assures that the solution leaves the obstacle precisely as a C 4 3 function. IV. Cavity and jet flows Within the theory of free boundary problems, the monotonicity formulae of Alt, Caffarelli, and Friedman [2] and Caffarelli, Jerison, and Kenig [6] are fundamental tools with a wide range of application. Motivated by physical questions involving free boundaries in non-euclidean ambients, in 2

3 collaboration with L. Zhang, we extend these monotonicity formulae to Riemannian manifolds, [Adv. Math. 226 (2011)] and [J. Geom. Anal. 21 (2011)]. This is, by no means, an easy task, as it involves a delicate perturbative analysis. For free boundary problems governed by nonlinear operators, a monotonicity formula cannot be established and the theory becomes more complex. In studies derived from the doctoral thesis of R. Leitão, published in [Rev. Mat. Iberoam. 31 (2015)] and [Ann. Inst. H. Poincare Anal. Non Linéaire 32 (2015)], we develop specific tools to attain sharp geometric estimates for free boundary variational problems governed by nonlinear degenerate elliptic operators of the p-laplacian type. Monotonicity formulae also cannot be proven if the linear problem is modeled within a discontinuous medium. Cavitation problems governed by bounded measurable coefficients was the subject of D. dos Prazeres Ph.D. thesis conducted under my supervision, [18]. Its main finding is that solutions are Lipschitz continuous along their free boundaries. This is an unexpected result as it would appear to conflict with the well-known fact that solutions to second-order elliptic PDE with measurable coefficients are merely Hölder continuous, as established in the theory of De Giorgi Nash Moser. The two-phase version of the problem is also relevant from the applied viewpoint, e.g. when one analyzes problems involving different constituents, such as ice melting in a heterogeneous fluid. In the Ph.D. thesis of M. Amaral, which underlies a joint study in [Comm. Math. Phys. 337 (2015)], we analyze such a two-phase transmission problem. III. CONTRIBUTIONS TO THE THEORY OF ELLIPTIC AND PARABOLIC PDE I. Nonvariational singular PDE Regularity estimates in the theory of diffusive models, i.e., problems ruled by second-order elliptic and parabolic equations, are among the greatest accomplishments of the modern analysis of PDE. They are present in a number of subfields, including differential geometry, numerical analysis, probability, harmonic analysis, mathematical physics, engineering mathematics, fluid dynamics, optimization, imaging process. The cornerstone chapters of this theory are the Schauder estimates of the 1930s, the De Giorgi Nash Moser theorem of the 1960s, and Krylov Safonov Harnack inequality which unlocked the theory of fully nonlinear PDEs of the 1980s. While these fundamental results enable us to address a large spectrum of problems, the mathematical analysis of equations with singular sources or potentials are considerably more complex, and thus present additional challenges. Until quite recently, for example, little, if anything, was known regarding fully nonlinear PDEs involving singular terms. In D. Araújo s doctoral thesis explicated in [Arch. Rational Mech. Anal. 209 (2013)], we developed a nonvariational approach to study equations of the form F(D 2 u) = u θ, 0 < θ < 1. Note that along the set where an existing solution vanishes (an a priori unknown region), the equation itself leads to Hessian blow-up. Hence, even in the linear case one cannot expect solutions to be classical. The study of this equation required the development of numerous analytic and geometric tools which permeate related subjects. It has been shown that solutions of this equation are of class C 1,α(θ) for an exponent 0 < α(θ) < 1, which depends solely on θ. In a previous study, jointly conducted with M. Montenegro [J. Funct. Anal. 259 (2010)], we attain gradient estimates for fully nonlinear equations, using Bernstein methods. To achieve that end, however, it was necessary to assume concavity of operator F. Anisotropic singular problems were subsequently examined in a jointly study with Montenegro and de Queiroz, [Math. Ann. 351 (2011)]. 3

4 II. Conformally invariant fully nonlinear equations The theory of isolated singularities of positive solutions of Yamabe-type equations has attracted the attention of researchers over recent decades. One of its principal motivations arises from the well-known conjecture of Yamabe: "In a Riemannian manifold (M, g), is there a metric ĝ which is pointwise conformal to g and has constant scalar curvature?". Jointly with Z. Han and Y. Li, [Invent. Math., 182, No. 3, (2010) ], we investigated he k-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric with applications to conformal geometry. In this study we prove that for every solution u of the equation it is possible to find a radial solution ū which is asymptotically close to the spherical average of u. The techniques used to establish our main result are based on a refinement of a fundamental work of Caffarelli, Gidas and Spruck [5]. At the heart of our approach lies a general asymptotic approximation result for solutions to certain ODE. An alternative approach based on linearized analysis, along the lines of Korevaar, Mazzeo, Pacard, and Schoen [16] was also carried out. Several interesting questions arise from our study, as well as the methods we developed, as reported therein. I am currently addressing problems of the Monge-Ampère type. In collaboration with L. Zhang, we have established a classification theorem when the forcing term is asymptotically close to a periodic function at infinity, in the spirit of Caffarelli and Li [7, 8]. III. Geometric regularity theory Recently, I have also directed considerably attention to the study of what has come to be known as geometric regularity theory, which fosters the development of effective tools for PDE analysis. A typical problem within this theory concerns sharp smoothness estimates for solutions of nonlinear problems. As an example, we cite the rekowned C 1,1/3 -regularity conjecture for best Lipschitz extension problem and the C p -regularity conjecture for functions whose p-laplacian is bounded. The Navier Stokes existence and smoothness problem is another example of these types of questions. Optimal estimates involving borderline integrability in sources is a first undertaking in this line of research. Sharp C 0,α estimates for degenerate elliptic and parabolic equations were established in [J. Math. Pures Appl. 99 (2013)] and in [Anal. PDE. 7 (2014)], the latter in collaboration with Urbano. The corresponding sharp estimates in the fully nonlinear setting were proven in [Arch. Rational Mech. Anal. 211 (2014)], which refines concepts derived from Caffarelli [4]. For nonvariational problems whose elliptic constants degenerate as u M, we proved, with Araújo and Ricarte, that solutions are precisely C 1, 1 M+1 regular, [Calc. Var. Partial Differential Equations 53 (2015)]. Sharp geometric growth estimates along the set of critical points were also investigated in this study. I have also used geometric tangential methods to establish sharp Hessian integrability estimates for solutions of uniformly elliptic equations F(x, D 2 u) = f (x) L p based solely on the behavior of the operator F at the ends of the space of symmetric matrices. The results of this research, conducted with E. Pimentel, a recent pos-doctorate student, have significant implications to the general theory of fully nonlinear equations. Among the most significant findings I contributed to the regularity theory of PDE to date are the improved Schauder estimates for problems with divergence and nondivergence structures. Since the 1930s, it has been a commonly accepted aphorism that the smoothness of the gradient and, respectively, the Hessian, of a solution could never be better than the continuity of the medium. However, in two articles [Math. Ann. 358 (2014)] and another to appear in Int. Math. Res. Not. I prove that higher estimates are possible at degenerate points. Surprisingly, such points constitute the set where the diffusion properties of the operator collapse. The proofs of this set of results are based on a new geometric approach to regularity theory, in which degenerate points are understood as as part of a nonphysical free boundary. Several applications of this new techniques are been undertaken. By way 4

5 of example, I note my study on fully nonlinear dead-core problems, to be published in Math. Ann. In this work I prove that viscosity solutions to nonlinear elliptic equations with merely measurable coefficients and µ-hölder continuous absorbing terms behave as C 2 1 µ -functions along the touching ground, regardless the low regularity of u within the non-coincidence set {u > 0}. In regard to degenerate elliptic equations of the p-laplace type, in a recent collaboration with Araújo and Urbano, we establish a new oscillation estimate which combined with geometric tangential methods and explicit estimates for p-harmonic functions from [17] yield a proof of the C p -regularity conjecture in dimension two. IV. RESEARCH AGENDA I hope to continue the development of geometric tangental methods for treating regularity issues in the theory of nonlinear PDE. Within the domain of prospective developments lies a unifying idea, viz., that regularity results have to be interpreted in an intrinsic geometric matter a sort of signature to each particular diffusive model. I intend to combine existing methods with new tools and insights based on geometric tangential analysis and free boundary problems to address the regularity issue in the theory of elliptic and parabolic PDE. Among the most outstanding problems awaiting a solution are the improved regularity estimates for the best Lipschitz extension problem. This problem comprises the analysis of viscosity solutions to highly degenerate elliptic equations ruled by the infinity Laplacian u := i u i j u j u. It has been observed by Evans and Smart [10, 11] that although stationary, the infinity Laplace equation can be regarded as a degenerate parabolic equation if the direction ν = D 2 udu is understood as time-like and the perpendicular directions, including that of the gradient, as space-like. This new perspective suggests that refitting tools from the theory of degenerate parabolic equations may be suitable to solve the problem. I plan to investigate it through a combination of those methods with new geometric insights. I am also interested in continuous differentiability properties for solutions to the inhomogeneous infinity Laplace equation, u = f (x). For instance, in principle, C 1 estimates for such an equation cannot be established by perturbative methods applied to Savin s result [19]. Nonetheless, I postulate that this new set of tools may overcome that difficulty, ultimately providing a way to address more general inhomogeneous problems. In the same line of inquiry, I am dedicated to proving the complete C p -regularity conjecture in higher dimensions. This problem inquires whether a W 1,p -function whose p-laplacian, p > 2, is bounded, i.e., p u = div( u p 2 u) = f (x) L, is locally of class C 1, 1 p 1. While it is well established that such functions are of class C 1,α for some 0 < α < 1, establishing the sharp estimate is a delicate matter. The approach proposed in [3] depends on explicit estimates for the homogeneous problem available only for two-dimensional equations. Based on recent developments, I expect that tools from the realm of algebraic geometry might hold the key to accessing the C p -regularity conjecture for dimensions larger than two. To explore this, collaborations from experts in the field of algebraic geometry would be welcomed. This new concept may well prove relevant to other problems, e.g., quantitative estimates for the general theory of fully nonlinear equations. I plan to investigate optimal W 2,δ -estimates for solutions of nondivergence form elliptic equations with the objective of proving that if C 1 i, j F(x, M) C, then δ = 2 C Another line of investigation that I intend to pursue concerns free boundary problems in discontinuous media. For cavitation problems, for instance, the objective is to attain, in a manner of speaking, a De Giorgi theorem for free boundaries. Regularity below the continuous threshold and the corresponding characterization of Reifenberg flat chord arc domains in terms of Poisson kernels have been studied by Kenig and Toro [12, 13, 14, 15]. In principle, if no continuity of the coefficients are assumed, the problem becomes quite unaccessible. However, there has been recent evidence that a 5

6 refined analysis is still possible even in discontinuous media [18]. A delicate step to this end will be to establish that the noncoincidence set is a NTA domain. The ultimate objective is to prove that the free boundary is uniformly rectifiable. Higher energy estimates are also expected for the the free boundary graph. The parabolic counterpart also warrants investigation. This constitutes the analysis of the singularly perturbed equation u t div(a i j (t, x)du) = β ɛ (u), where β ɛ is an approximation of the Dirac delta in L 1. I also plan to develop the fully nonlinear analogues in the one- and two-phase cases. Of late, an emerging line of research in which diffusive processes admit a continuous spectrum of influence has attracted considerable attention. Such considerations lead to the study of elliptic operators with fractional diffusions. Extensive literature has been published on the topic, but several important issues remain unresolved. Among them, I note fractional two-phase problems involving one-sided penalization. This problem is challenging because it is expected that solutions behave as in the cavity problem in one phase and as the obstacle problem in the other. Hence we anticipate that a monotone free boundary condition u + ν = G(u ν ) will govern the flux balance in a rather interesting manner. REFERENCES [1] Alt, H.; Caffarelli, L. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), [2] Alt, H.; Caffarelli, L. ; Friedman, A. Variational problems with two phases and their free boundaries. Trans. Amer. Math. Soc. 282 (1984), no. 2, [3] Araujo, A.; Teixeira, E.; Urbano, J.M. A proof of the C p -regularity conjecture in the plane. Preprint. [4] Caffarelli, L. Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), [5] Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local be- havior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, [6] Caffarelli, L.; Jerison, D.; Kenig, C. Some new monotonicity theorems with applications to free boundary problems. Ann. of Math. (2) 155 (2002), no. 2, [7] Caffarelli, L.; Li, Y. A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 1, [8] Caffarelli, L.; Li, Y. An extension to a theorem of Jörgens, Calabi, and Pogorelov. Comm. Pure Appl. Math. 56 (2003), no. 5, [9] Evans, L.; Savin, O. C 1,α regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations 32 (2008), [10] Evans, L. C.; Smart, C. Adjoint methods for the infinity Laplacian partial differential equation. Arch. Ration. Mech. Anal. 201 (2011), no. 1, [11] Evans, L. C.; Smart, C. Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Diffierential Equations 42 (2011), [12] Kenig, C.; Toro, T. Harmonic measure on locally flat domains. Duke Math. J. 87 (1997), no. 3,

7 [13] Kenig, C.; Toro, T. Free boundary regularity for harmonic measures and Poisson kernels Ann. of Math. (2) 150 (1999), [14] Kenig, C.; Toro, T. Poisson kernel characterization of Reifenberg flat chord arc domains. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 3, [15] Kenig, C.; Toro, T. Free boundary regularity below the continuous threshold: 2-phase problems. J. Reine Angew. Math. 596 (2006), [16] N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math. 135 (1999), [17] Iwaniec, T.; Manfredi, J. Regularity of p-harmonic functions on the plane, Rev. Mat. Iberoamericana 5 (1989), [18] dos Prazeres, D.; Teixeira, E. Cavity problems in discontinuous media. Preprint. [19] Savin, O. C 1 regularity for infinity harmonic functions in two dimensions, Arch. Rational Mech. Anal. 76 (2005),