Many phenomena in the physical sciences are governed by nonlinear partial differential equations (NLPDE). Almost all the problems in this proposal involve "peaks of concentration"; they correspond to small regions in space (often assimilated to points or lines) where some of the variables can take extremely high values. Such peaking zones can be part of the data or part of the unknown. For example, in nuclear physics, positive nuclei are fixed zones of high density surrounded by a diffuse cloud of electrons floating around them. A surprising mathematical discovery is that some natural NLPDE admit no solution when high-density data are concentrated in regions that are "too small." In mathematical language, this can be expressed by saying that some measures (e.g., Dirac masses) are not admissible data. The principal investigator proposes to classify all admissible measures for a large class of NLPDE. In other problems, the zones of high concentration are part of the unknown. Singularities may appear when a "mild" external field is applied to the system. The examples of this phenomenon that are relevant to the current project arise in the physics of liquid crystals and superconductors. In this project, the principal investigator will continue his research in several directions, addressing such issues as the following: What kind of external action is required to produce singularities? Describe the nature, the strength, and the location of singularities?
In the real world, one often encounters phenomena of extreme intensity, which appear in small regions of space or persist only during a small time interval. A short list of examples includes the following: vortices (similar to tornadoes) in fluid mechanics and superconductors, fractures in solid mechanics, self-focusing beams in nonlinear optics (e.g., in lasers), defects in liquid crystals, black holes in astrophysics. One of the project?s goals is to describe all possible singular behaviors for a large class of nonlinear models. It is important to understand what causes such "blow-up" phenomena, in order either to avoid them or to enhance them. Mathematically similar problems occur not only in physics but in many other areas of mathematics.
Highlights of main accomplishments: 1) Hilbert's nineteenth problem (from his celebrated list compiled in 1900) was solved independently by E. DeGiorgi and J. Nash in 1957. The main assumption in DeGiorgi’s fundamental theorem (u has square integrable first derivatives) seemed too strong, but Serrin showed in 1964 that, in fact, it could not be weakened. Serrin also conjectured that under the natural weak condition on the solution (u has integrable first derivatives), the conclusion of DeGiorgi’s theorem would still hold under a stronger condition on the coefficients of the differential operator (Hölder continuity). In 2008 I established Serrin's conjecture which had been open since 1964 (see [Br]). 2) The concept of Jacobian determinant plays a central role in Calculus. It occurs prominently e.g. when performing a change of variables F in multidimensional integrals. The standard theory requires that F should be continuously differentiable. We are able give a "robust" and useful definition of the Jacobian determinant for a vast array of irregular maps. In particular we identify the largest such class in the framework of maps having only fractional derivatives. Our results have appeared in two of the most eminent mathematical journals (see [Br-Ngu-1], [Br-Ngu-2]). 3) Maps with values into the unit circle are ubiquitous in physical problems. A standard example is the orientation of a compass needle viewed as a function of a point varying on the surface of the earth. The orientation varies regularly except at the North Pole and the South Pole. Such points are called the singularities of the map. Mathematicians work with Sobolev maps (rather than smooth maps) precisely to allow maps with singularities. They occur naturally in many physical phenomena, e.g. superconductors. I have investigated thoroughly Sobolev maps with values into the circle and I am preparing a Research Monograph on this subject (see [Br-Mir]). Here is a typical subject I have studied. The position of a point on the unit circle is determined by an angle (called the lifting of the original map). Any map admits plenty of liftings (since the angle is measured modulo 2π). "Optimal" liftings, in the sense that they have least total variation, play a distinguished role. After much effort we have been able to classify all optimal liftings: there is a one-to-one correspondence between optimal liftings of a given map and minimal surfaces spanned by its singularities. 4) The need for efficient image restoration methods has grown with the massive production of digital images often taken, or transmitted, in poor conditions. Likewise, to achieve the best possible diagnosis it is important that medical images be sharp and free of noise. Blurred and distorted images need to be "filtered" before one can extract reliable information. Over the past twenty years a vast array of filters has been devised. In June 2011, while attending a conference on this subject--- and after discussions with experts--- I realized that some tools I developed earlier (in connection with questions of total variation described in section 3) could be useful in Image Processing. My contributions are twofold. Firstly, I discovered unexpected connections between various filters which were seemingly unrelated. Secondly, I proposed new filters which may lead to new algorithms used in practice (see [Br-Ngu-3]). 5) My PhD students Hernan Castro and Hui Wang, supported through this Grant, have discovered a beautiful Hardy-type inequality involving the integral of second-order derivatives. Such a result is totally original and unexpected because similar inequalities involving first-order derivatives are not true (see [Cas-Dav-Wang]). References [Br] H. Brezis, On a conjecture of J. Serrin, Rend Lincei Mat. Appl., 19 (2008), 335—338. [Br-Mir] H. Brezis and P. Mironescu, "Sobolev Maps with Values into the Circle---Analytical, Geometrical and Topological Aspects", Birkhäuser Research Monograph, (in preparation). [Br-Ngu-1] H. Brezis and H.-M. Nguyen, On the distributional Jacobian of maps from SN into SN in fractional Sobolev and Holder space, Annals of Math., 173 (2011), 1141-1183. [Br-Ngu-2] H. Brezis and H.-M. Nguyen, The Jacobian determinant revisited, Invent. Math., 185 (2011), 17-54. [Br-Ngu-3] H. Brezis and H.-M. Nguyen, Nonlocal functionals related to the total variation and applications in Image Processing, (to appear). [Cas-Dav-Wang] H. Castro, J. Davila and H. Wang, A Hardy type inequality for W m,1(Ω) functions, J. Eur. Math. Soc 15 (2013), 145-155.
