Sl s2 be a c2 immersion with length l and total curvature k. To the joy of analysts everywhere, we can rephrase this theorem as an inequality. Journal of mathematical inequalities rg journal impact. A popular isoperimetric inequality is known as the classical isoperimetrical inequality. Mathematical physics, spectral theory and stochastic analysis, 1106. Szeg o, isoperimetric inequalities in mathematical physics, annals of mathematics studies, no. The calculus of variations evolved from attempts to solve this problem and the. Isoperimetric inequalities in mathematical physics annals of. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Energy inequalities in interacting quantum field theories. Isoperimetric inequalities in mathematical physics anisa m. The case of equality follows readily, since in particular, equality holds in our use of the isomperimetric inequality on t. By means of an iteration process and symmetrization methods the solution in an arbitrary domain is compared with the one for the sphere of the same volume.
In this paper we are interested in isoperimetric inequalities of the logarithmic potential l. We present here an isoperimetric inequality for sets contained in a convex cone. Equality case for an elliptic area condenser inequality. Isoperimetric inequalities for positive solution of plaplacian huaxiang hu andqiuyi dai abstract.
Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. The book description for the forthcoming isoperimetric inequalities in mathematical physics. H p,l p type inequalities of maximal operators of vilenkinnorlund means with. Isoperimetric inequalities in mathematical physics gyorgy. Advances in mathematics 37, 8398 1980 isoperimetric inequalities on curved surfaces isaac chavel and edgar a. Lectures on isoperimetric and isocapacitary inequalities in. Isoperimetric inequality encyclopedia of mathematics. Related books essential discrete mathematics for computer science harry lewis and rachel zax. Isoperimetric inequalities for the logarithmic potential.
Department of mathematical physics, national university of ireland, maynooth, ireland and dublin institute for advanced studies, 10 burlington rd. In this paper, weshow that the socalled sneakout principle for discrete inequalities is valid also on a general time scale. Isoperimetric inequalities in mathematical physics book. The uncertainty principle in terms of isoperimetric. These conemanifolds include orientable orbifolds, and singular geometric structures on 3manifolds studied by w. The interplay between physics and geometry gives to geometric inequalities in general relativity their distinguished character. Research article on isoperimetric inequalities in minkowski spaces horst martini1 and zokhrab mustafaev2 1 faculty of mathematics, university of technology chemnitz, 09107 chemnitz, germany 2 department of mathematics, university of houstonclear lake, houston, tx 77058, usa.
Isoperimetric inequalities in mathematical physics princeton u. The stong isoperimetric inequality of bonnesen andrejs treibergs university of utah among all simple closed curves in the plane with a. Isoperimetric inequalities in mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Thermodynamic volumes and isoperimetric inequalities forde sitterblackholes brian p. Equality case for an elliptic area condenser inequality and a related schwarz type lemma volume 63 issue 1 georgios kelgiannis. Prices in gbp apply to orders placed in great britain only. A geometrical isoperimetric inequality and applications to. Pdf the application of isoperimetric inequalities for. Prices in represent the retail prices valid in germany unless otherwise indicated.
The inequalities proved provide upper and lower bounds for the. Isoperimetric literally means having the same perimeter. We will present a survey on isoperimetric inequalities in real. Boolean properties and belllike inequalities of numerical events. Fourth summer school in analysis and mathematical physics. Other readers will always be interested in your opinion of the books youve read.
This paper studies eigenvalue bounds and isoperimetric inequalities for riemannian spaces with cone type singularities along a codimension2 subcomplex. Buy isoperimetric inequalities in mathematical physics annals of mathematics studies by george polya, gabor szego isbn. These inequalities involve a symmetrization process very closely related to steiner symmetrization used in the theory of isoperimetric inequalities. Mar 28, 2010 applications of such inequalities can be found in stochastic geometry, functional analysis, fourier analysis, mathematical physics, discrete geometry, integral geometry, and various further mathematical disciplines. In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. Numerous and frequentlyupdated resource results are available from this search. The isoperimetric inequality project euclid mathematics. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of errorcorrecting codes. Also given are necessary and sufficient conditions for the equivalence between the extremality of some sets in the isoperimetric problem and the validity of some analytic inequalities. Isoperimetric inequalities and their applications siam. The abstract concepts it presents may seem inapplicable to realworld problems to the layman. Isoperimetric inequalities in mathematical physics szego, gabor, polya, george. New inequalities for certain greens functions are given.
On isoperimetric inequalities in minkowski spaces journal. Venants conjectures to bounded simply connected domains on curved 2dimensional riemannian manifolds. Jacqueslouis lions, 1928 2001 jacqueslouis lions was born in 1928 in grasse in the maritime alps. We will present a survey on isoperimetric inequalities in real, finitedimensional banach spaces, also called minkowski spaces. Those are simply variational problems with constraints, whose name derives from the fact that inequality 1 corresponds to the first example of such a problem. Am27, volume 27 annals of mathematics studies on free shipping on qualified orders. Isoperimetric and other inequalities in the theory of neutron. Isoperimetric inequalities for positive solution of p. Descargar libro isoperimetric inequalities in mathematical physics.
In this paper, we prove some isoperimetric inequalities and give a explicit bound for the positive solution of plaplacian. Existence theorems and a priori bounds for a class of nonlinear parabolic equations are established. Department of mathematics and program in applied mathematics, university of arizona, tucson, arizona 85721. The author discusses inequalities in euclidean and riemannian geometry, methods of classical differential geometry and elementary modern geometric measure, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on riemannian manifolds. Bulletin of the american mathematical society project euclid. It is shown that among all domains of given volume the sphere is the least stable. The rst and best known isoperimetric inequality is the classical isoperimetric inequality a l2 4 relating the area aenclosed by a planar closed curve of perimeter l. He attended the university of paris and received his doctorate in science from there in 1954. Much attention is devoted to probability distributions on the real line, the normalized lebesgue measure on the euclidean spheres, and the canonical gaussian. Discrete isoperimetric inequalities ucsd mathematics. Some isoperimetric inequalities in the torsion problem for multiply connected regions, studies in mathematical analysis and related topics. Pdf advances in mathematical inequalities by shigeru furuichi free downlaod publisher. In the absence of any restriction on shape, the curve is a circle. Isoperimetric inequalities in mathematical physics.
Some isoperimetric and other inequalities occurring in the one. The mathematics of inequality new statesman hussonet free. Isoperimetric inequalities in the physics of black holes article pdf available in acta physica polonica series b 22. Chorwadwala a typical isoperimetric problem is to enclose a given area a with a shortest possible curve. Isoperimetric inequalities with practical applications. Inequalities for means of chords, with application to. An isoperimetric inequality for extremal sobolev functions. Introduction union college math department home page. We consider a homogeneous membrane covering a region r2. Gerhard huisken max planck institute of gravitational physics albert einstein institute in postdam germany a tea will precede lectures at 3. The isoperimetric inequality, the mathematical statememt of this fact, is done in two parts.
Inequalities and uncertainty principles journal of mathematical physics 19, 461. Annals of mathematics studies, 27, princeton university press. Isoperimetric and other inequalities in the theory of. The application of isoperimetric inequalities for nonlinear eigenvalue problems. The classical isoperimetric theorem asserts that in the euclidean plane the unique solution is a circle. Harrell c, michael loss a department of theoretical physics, nuclear physics institute, academy of sciences, 25068 re. Mathematical inequalities volume 10, number 2 2016, 393403 doi. In this paper we introduce the concept of area functions for a given function. Am27, volume 27 by georg polya, 9780691079882, available at book depository with free delivery worldwide.
Isoperimetric inequalities for a class of nonlinear parabolic. Rayleighs conjecture on the principal frequency of the clamped. In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties. Lectures on isoperimetric and isocapacitary inequalities in the theory of sobolev spaces vladimir mazya abstract.
They may be interpreted physically in many ways, for example, as applying to the quantum mechanical motion of a particle in a potential or to diffusion in the presence of absorbers. Dec 22, 2004 some isoperimetric and other inequalities occurring in the one. Feldman the city college and the graduate school of the city university of new york, new york, new york in this paper we extend the solutions of lord rayleighs and st. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. This paper aims to demystify isoperimetric inequalities. Isoperimetric inequalities for eigenvalues of the laplacian.
Isoperimetric inequalities on curved surfaces sciencedirect. Isoperimetric inequalities have played an important role in mathematics since the times of ancient greece. This proof may be more technical and not as elegant as other proofs but it can be given with basic mathematical. Lecture 21 inequalities arithmetic geometric mean inequality. The class of isoperimetric inequalities is enriched by mathematical physics, the theory of functions of a complex variable, functional analysis, the theory of approximations of functions, and the calculus of variations.
Requiring only of a basic course in differential geometry, this. If a is regularly homotopic to a circle traversed once then. Research article on isoperimetric inequalities in minkowski. There are also the isoperimetric inequalities of mathematical physics. The isoperimetric problem mathematical association of. These inequalities have become powerful tools in modern mathematics. Isoperimetric problem, in mathematics, the determination of the shape of the closed plane curve having a given length and enclosing the maximum area. An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in gauss space bobkov, s.
Old and new authors results on equivalence of various isoperimetric and isocapacitary inequalities, on one hand, and sobolevs type imbedding and compactness theorems, on the other hand, are described. Inequalities for means of chords, with application to isoperimetric problems pavel exnera,b, evans m. In their famous book isoperimetric inequalities in mathematical physics, polya and szego extended this notion to include inequalities for domain functionals, provided that the equality sign is attained for some domain or in the limit as the domain degenerates 15. The quantities involved in these inequalities all refer to bare solids with isotropic scattering and are. We will then give a geometric proof by jakob steiner to show an easily comprehensive approach, followed by a rigorous proof by erhard schmidt.
Buy isoperimetric inequalities in mathematical physics. Thermodynamic volumes and isoperimetric inequalities forde. A survey on the kreinvon neumann extension, the corresponding abstract buckling problem, and weyltype spectral asymptotics for perturbed krein laplacians in nonsmooth domains. Recent reports on mathematical physics articles elsevier. Pdf isoperimetric inequalities in the physics of black holes.
Isoperimetric inequalities in riemannian geometry are noticeably more complex. Others may be found in a recent paper of the author 4 on bonnesen inequalities and in the book of santalo 4 on integral geometry and geometric probability. Full text views reflects the number of pdf downloads, pdfs. In particular, we strengthen liebs triple matrix inequality. The description for this book, isoperimetric inequalities in mathematical physics. Eigenvalue estimates and isoperimetric inequalities for. Some applications to symmetrization problems and sobolev inequalities are also indicated. A geometrical isoperimetric inequality and applications to problems of mathematical physics. Isoperimetric inequalities in mathematical physics szego. The main goal of the articles is to link the basic knowledge of a graduate student in mathematics with three current research topics in mathematical physics. The application of isoperimetric inequalities for nonlinear. There are also isoperimetric inequalities of mathematical physics. Everyday low prices and free delivery on eligible orders.
1592 1364 983 584 1504 13 711 406 612 1438 1201 36 1293 787 571 278 1297 270 714 676 1333 1071 161 1404 1160 817 78 1494 840 654 758 504 828 1037 947 1396 279 1283 408 721 1488 709 189 452 1163