harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

power concavity on nonlinear parabolic flows our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p laplace equation called power concavity and that the power is determined by the homogeneity of the parabolic operator in the parabolic p laplace equation for the density u the concavity of u p 2 p is considered which indicates why the log concavity has been considered in heat flow p 2 in addition the long time existence of the classical solution of the parabolic p laplacian equation can be obtained if the initial smooth data has concavity and a nondegenerate gradient along the initial boundary 2004 wiley periodicals inc

power concavity in weakly coupled elliptic and parabolic systems abstract in this paper we start the investigation of concavity properties of solutions to systems of pde s in convex domains in particular we prove that suitable powers of solutions to some weakly coupled elliptic and parabolic systems are concave

boundary behavior of the fast diffusion equation the fast diffusion equation avm vt 0 m 1 is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established in this paper we are going to obtain a lipschitz growth rate of the solution at the boundary of nl and estimate that in terms of the various data

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

empirical geometry of multivariate data a deconvolution approach let y j j 1 n be independent observations in r m m 1 with common distribution q suppose that y j x j j j 1 n where x j j j 1 n are independent x j j 1 n have common distribution p and j j 1 n have common distribution so that q p the problem is to recover hidden geometric structure of the support of p based on the independent observations y j assuming that the distribution of the errors is known deconvolution statistical estimates of the fractal dimension and the hierarchical cluster tree of the support that converge with exponential rates are suggested moreover the exponential rates of convergence hold for adaptive versions of the estimates even in the case of normal noise j with unknown covariance in the case of the dimension estimation though the exponential rate holds only when the set of all possible values of the dimension is finite e g when the dimension is known to be integer if this set is infinite the optimal convergence rate of the estimator becomes very slow typically logarithmic even when there is no noise in the observations

distance preserving and distance increasing mappings from ternary vectors to permutations permutation arrays have found applications in powerline communication one construction method for permutation arrays is to map good codes to permutations using a distance preserving mappings dpm dpms are mappings from the set of all q ary vectors of a fixed length to the set of permutations of some fixed length the same or longer such that every two distinct vectors are mapped to permutations with the same or larger hamming distance than that of the vectors a dpm is called distance increasing dim if the distances are strictly increased except when the two vectors are equal in this correspondence we propose constructions of dpms and dims from ternary vectors the constructed dpms and dims improve many lower bounds on the maximal size of permutation arrays

large time geometrical properties of solutions of the barenblatt equation of elasto plastic filtration abstract we study the geometric behavior for large times of the solutions of the following equation u t u t u 0 1 posed in the whole space r n for n 1 when the initial data are nonnegative continuous and compactly supported we prove that after a finite time log u becomes a concave function in the space variable and converges to all orders of differentiability to a certain parabolic shape so called barenblatt type profile which was described in kamin et al 1991 20 extensions to more general fully nonlinear equations are considered

strict convexity of level sets of solutions of some nonlinear elliptic equations in this paper we study the convexity of the level sets of solutions of the problem where f is a suitable function with subcritical or critical growth under some assumptions on the gauss curvature of we prove that the level sets of the solution of 0 1 are strictly convex

geometric properties of the gelfand problem through a parabolic approach abstract we consider the asymptotic profiles of the nonlinear parabolic flows e u t u e u to show the geometric properties of minimal solutions of the following elliptic nonlinear eigenvalue problems known as the gelfand problem e 0 0 in 0 on posed in a strictly convex domain r n in this work we show that there is a strictly increasing function f s such that f 1 x is convex for 0 i e we prove that level set of is convex moreover we also present the boundary condition of which guarantees the f convexity of solution

on the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity 497 abstract we investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial boundary value problems with convex or concave nonlinearity in the convex case f 0 0 implies instability in the concave case f 0 0 implies stability we also discuss the necessity of the sign condition on f 0

convexity and all time c regularity of the interface in flame propagation we consider the following one phase free boundary problem find u such that u 0 and with qt n 0 t under the condition that o is convex and log uo is concave we show that the convexity of t and the concavity of log u t are preserved under the flow for 0 t t as long as t and u on t are smooth as a consequence we show the existence of a smooth up to the interface solution on 0 t tc with tc denoting the extinction time of t we also provide a new proof of a short time existence with c2 initial data on the general domain 2002 john wiley sons inc

long time existence of smooth solution for the porous medium equation in a bounded domain in this paper we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 ut umin 0 u x 0 u0 0in u x t 0for x where m 1 is the permeability the proof is based on the short time existence of cs2 smooth solution the global cs1 estimate the holder estimate of divergence type degenerate equation with measurable coefficients and cs1 estimate of mixed type equation with lipschitz coefficients

large time geometric properties of solutions of the evolution p laplacian equation abstract we establish the behavior of the solutions of the degenerate parabolic equation u t u p 2 u p 2 posed in the whole space with nonnegative continuous and compactly supported initial data we prove a nonlinear concavity estimate for the pressure away from the maximum point the estimate has important geometric consequences it implies that the support of the solution becomes convex for large times and converges to a ball in dimension one we know also that the pressure itself eventually becomes p concave in several dimensions we prove concavity but for a small neighborhood of the maximum point

parabolic power concavity and parabolic boundary value problems this paper is concerned with power concavity properties of the solution to the parabolic boundary value problem begin aligned p quad left begin array l quad l partial t u vardelta u f x t u nabla u text in quad varomega times 0 infty u x t 0 text on quad partial varomega times 0 infty u x 0 0 text in quad varomega end array right end aligned where varomega is a bounded convex domain in mathbf r n and f is a nonnegative continuous function in varomega times 0 infty times mathbf r times mathbf r n we give a sufficient condition for the solution of p to be parabolically power concave in overline varomega times 0 infty

distance increasing mappings from binary vectors to permutations mappings from the set of binary vectors of a fixed length to the set of permutations of the same length that strictly increase hamming distances except when that is obviously not possible are useful for the construction of permutation codes in this correspondence we propose recursive and explicit constructions of such mappings some comparisons show that the new mappings have better distance expansion distributions than other known distance preserving mappings dpms we also give some examples to illustrate the applications of these mappings to permutation arrays pas

smooth solution for the porous medium equation in a bounded domain abstract in this paper we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 u t u m m 1 with zero boundary condition on the fixed boundary the flux u m is nonzero while the gradient of pressure u m 1 is zero as a consequence the parabolic equation above becomes degenerate the proof is based on a definition of weighted space c s 2 corresponding to the given degeneracy and the schauder estimates in its linearized equation 0 2 w t x n a i j x d i j w in c s 2 for 0 m 1 m 1

distance increasing mappings from binary vectors to permutations that increase hamming distances by at least two in this correspondence for any k ges 2 we first propose two constructions of n k distance increasing mappings dims from the set of binary vectors of length n to the set of permutations of the same length that strictly increase the hamming distance by at least k except when it is obviously not possible next we prove that for any k ges 2 there is a smallest positive integer nk such that an n k dim can be constructed for any n ges nk an explicit upper bound on nk is also given

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

a generalization of caffarelli s contraction theorem via reverse heat flow a theorem of l caffarelli implies the existence of a map pushing forward a source gaussian measure to a target measure which is more log concave than the source one which contracts euclidean distance in fact caffarelli showed that the optimal transport brenier map topt is a contraction in this case we generalize this result to more general source and target measures using a condition on the third derivative of the potential by providing two different proofs the first uses a map t whose inverse is constructed as a flow along an advection field associated to an appropriate heat diffusion process the contraction property is then reduced to showing that log concavity is preserved along the corresponding diffusion semi group by using a maximum principle for parabolic pde in particular caffarelli s original result immediately follows by using the ornstein uhlenbeck process and the prekopa leindler theorem the second uses the map topt by generalizing caffarelli s argument employing in addition further results of caffarelli as applications we obtain new correlation and isoperimetric inequalities

asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems abstract we study the fully nonlinear parabolic equation f d 2 u m u t 0 in 0 m 1 with the dirichlet boundary condition and positive initial data in a smooth bounded domain r n provided that the operator f is uniformly elliptic and positively homogeneous of order one we prove that the renormalized limit of parabolic flow u x t as t is the corresponding positive eigenfunction which solves f d 2 p 0 in where 0 p 1 m 1 and 0 is the corresponding eigenvalue we also show that some geometric property of the positive initial data is preserved by the parabolic flow under the additional assumptions that is convex and f is concave as a consequence the positive eigenfunction has such geometric property that is log is concave in the case p 1 and 1 p 2 is concave for 0 p 1