A Dimension Free Inequality And Its Application In Partial Differential Equations

Inequalities are important tools when people are trying to solve many mathematical problems, for instance, the Jensen inequality, Holder inequality, Minkowski inequality and Sobolev inequalities play a fundamental role in mathematical analysis. They are in particular indispensable tools in the study of partial differential equations. A feature of Sobolev inequalities is that Sobolev constants depend on the dimension of the Euclidean spaces.In the theory of quantum field, due to the occurrence of infinitely many dimensions in many problens, it is extremely important and useful to have estimates which is dimension free, such as the logarithmic Sobolev inequality. In this paper, as an elementary attempt, we establish the following inequality in an arbitrary Hilbert space H:Where ?>-1/2,x,y?H are arbitrary two points, andc₁,c₂,are two positive constants depending only on ?.Our inequality originates from the study of regularity theory of p-harmonic mappings. p-harmonic mappings are a natural extension of harmonic mappings, and the. Associated Euler-Lagrange equations are also the most similar second order PDE to that of harmonic mappings. It is expected that p-harmonic mappings have quite similar properties to that harmonic mappings, and thus gained extensive research in the literature.In the papers Giaquinta-Modica [27] and Acerbi-Fuseo [32], to obtain higher order regularity of minimizers of a class of p-Laplace type nonlinear functional between Euclidean spaces in the regularity theory of PDE of second order. It is the monotonicity of p-Laplace type operators that makes this method applicable. To this end, they proved that inequality ?*? holds for all x,y?Rⁿ, and for some constants c₁,c₂, which depends not only on ? but also on the dimension n.Thus,our result can be viewed as an improvement and an extension of their inequalities.