local behavior of solutions of quasilinear elliptic equations with general structure this paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form diva x u vu b x u vu where a and b are borel measurable are solutions to the equation diva x u vu b x u vu u for some nonnegative radon measure i among other things it is shown that if u is a holder continuous solution to this equation then the measure u satisfies the growth property j b x r 1 is given by the structure of the differential operator conversely if u is assumed to satisfy this growth condition then it is shown that u satisfies a harnack type inequality thus proving that u is locally bounded under the additional assumption that a is strongly monotonic it is shown that u is holder continuous

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

extremal solutions of p laplacian problems in d1 p rn via wolff potential estimates abstract we prove the existence of smallest positive and greatest negative solutions in d 1 p r n of the p laplacian problem of logistic type in r n with 1 p n u d 1 p r n p u a x u p 2 u g u and consider its extended version with a harvesting term h as well unlike papers on the topic here we only assume a superlinear growth for the nonlinearity g r r at zero and at infinity i e even supercritical growth is allowed and moreover no monotonicity for s g s s p 1 is required we prove a vazquez type maximum principle for nonnegative d 1 p r n solutions of some p laplacian problem and establish a sub supersolution method in d 1 p r n which along with wolff potential estimates for superharmonic functions will be used as our main tools

singular integral equation analysis of flange mounted rectangular waveguide radiators the authors analyze rigorously single and twin radiating waveguides by means of a singular integral equation approach which includes the edge condition on the apertures from the start the above inclusion has an important bearing on convergence and matrix size enabling highly accurate solutions to be obtained with few test functions typically two to four this becomes particularly important in the case of multiple coupled radiators where matrix sizes increase as the product of the number of radiators times the number of test functions theoretical and experimental results are compared for the two important cases of two radiators with parallel narrow walls and with parallel broad walls and are found to be in very good agreement

gradient estimates for problems with orlicz growth abstract we study a general nonlinear elliptic equation in the orlicz setting with data not belonging to the dual of the energy space we provide several lorentz type and morrey type estimates for the gradients of solutions under various conditions on the data

the dirichlet problem for p harmonic functions with respect to the mazurkiewicz boundary and new capacities in this paper we develop the perron method for solving the dirichlet problem for the analog of the p laplacian i e for p harmonic functions with mazurkiewicz boundary values the setting conside

isolated singularities of solutions to quasi linear elliptic equations with absorption we study the problem of removability of isolated singularities for a general second order quasi linear equation in divergence form diva x u u a0 x u g x u 0 in a punctured domain 0 where is a domain in rn n 3 the model example is the equation pu gu u p 2 u u q 1 0 q p 1 0 p n assuming that the lower order terms satisfy certain non linear kato type conditions we prove that for q n p 1 n p all point singularities of the above equation are removable thus extending the seminal result of brezis and veron

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

existence and asymptotic behavior of nodal solutions for semilinear elliptic equations abstract the elliptic equation u u 0 in r n is discussed in the case where u u p 1 u u 1 u q 1 u u p n 2 n 2 q and n 3 it is shown that any radially symmetric solution behaves as x like one of three types i c x n 2 for some c 0 ii c x 2 q 1 iii c x 2 q 1 where c 2 q 1 1 n 2 2 q 1 1 q 1 0 it is further proved that for any k 0 there exist at least three radially symmetric solutions which have exactly k zeros in ihe interval 0 x

structure of positive radial solutions of semilinear elliptic equations abstract we study the positive radial solutions of a semilinear elliptic equation u f u 0 wheref u has a supercritical growth order for smallu 0 and a subcritical growth order for largeu by showing the uniqueness of positive solutions behaving likeo x 2 n at infinity we give an almost complete description for the structure of positive radial solutions as a consequence we also prove the uniqueness of positive solutions of the nonlinear dirichlet problem for the equation in a finite ball

minima of quasisuperminimizers abstract we show that unlike minima of superharmonic functions which are again superharmonic the same property fails for q quasisuperminimizers more precisely if u i is a q i quasisuperminimizer i 1 2 where 1 q 1 q 2 then u min u 1 u 2 is a q quasisuperminimizer but there is an increase in the optimal quasisuperminimizing constant q we provide the first examples of this phenomenon i e that q q 2 in addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to kinnunen and martio moreover our lower and upper bounds turn out to be quite close we also study a similar phenomenon in pasting lemmas for quasisuperminimizers where q q 1 q 2 turns out to be optimal and provide results on exact quasiminimizing constants of piecewise linear functions on the real line which can serve as approximations of more general quasiminimizers

sharp exponents and a wiener type condition for boundary regularity of quasiminimizers abstract we obtain a sufficient condition for boundary regularity of quasiminimizers of the p energy integral in terms of a wiener type sum the exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant we also show by an example that the exponent is sharp in a certain sense

existence of minimal solutions to quasilinear elliptic equations with several sub natural growth terms abstract we study the existence of positive solutions to quasilinear elliptic equations of the type p u u q in r n in the sub natural growth case 0 q p 1 where p u u p 2 u is the p laplacian with 1 p n and and are nonnegative radon measures on r n we construct minimal generalized solutions under certain generalized energy conditions on and to prove this we give new estimates for interaction between measures we also construct solutions to equations with several sub natural growth terms using the same methods

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

the dirichlet problem for p harmonic functions on metric spaces we study the dirichlet problem for p harmonic functions and p energy minimizers in bounded domains in proper pathconnected metric measure spaces equipped with a doubling measure and supporting a poincare inequality the dirichlet problem has previously been solved for sobolev type boundary data and we extend this result and solve the problem for all continuous boundary data we study the regularity of boundary points and prove the kellogg property i e that the set of irregular boundary points has zero p capacity we also construct p capacitary p singular and p harmonic measures on the boundary we show that they are all absolutely continuous with respect to the p capacity for p 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide we give an integral representation for the solution to the dirichlet problem when p 2 enabling us to extend the solvability of the problem to l 1 boundary data in this case moreover we give a trace result for newtonian functions when p 2 finally we give an estimate for the hausdorff dimension of the boundary of a bounded domain in ahlfors q regular spaces less

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

regularity symmetry and uniqueness of some integral type quasilinear equations abstract we study integral equations corresponding to some quasilinear equations with nonlinearities of lane emden and hartree type regularity symmetry and uniqueness of these equations are considered we obtain the uniqueness of the ground state of h 1 critical hartree equation and extend the moving plane method of integral equation in w chen c li b ou classification of solutions for an integral equation comm pure appl math lix 2006 0330 0343 w chen c li b ou classification of solutions for a system of integral equations comm partial differential equations 30 1 3 2005 59 65 to some integral equations corresponding to the p laplace equation we use ideas from the potential theories for the p laplace equations and hessian equations

generalized network formulation application to flange mounted radiating waveguides electromagnetic linear field problems can be rigorously and systematically formulated in terms of linear networks use of the equivalence theorem permits the decomposition of the computational domain into sub domains with common interfaces in each sub domain the basic building blocks for a linear network representation are provided by tellegen s theorem and by the two alternative representations of the green s function tellegen s theorem provides the basis for expressing field at interfaces in a consistent way the eigenfunction expansion of the green s function provides a resonant type equivalent circuit useful for expressing the field inside a sub domain while the green s function direct solution emphasizes a transmission line network in particular a spherical transmission line provides the network representation of radiation problems in this paper the generalized network formulation is applied to the problem of radiation from flange mounted waveguides while this problem has been already solved with a variety of approaches the advantage of the generalized network formulation is that while comprising the previously developed approaches it also allows rigorous derivation of network representations in particular it is shown that by expressing the field in the half space in terms of spherical waves the physically matched solutions provide in some cases advantages for numerical computations and for network representations

potential estimates of superquadratic elliptic systems with vmo coefficients in reifenberg domains abstract the purpose of this paper is to study an inhomogeneous dirichlet problem of a superquadratic nonlinear elliptic system with vmo coefficients in reifenberg flat domains we derive pointwise potential estimates for weak solution to the dirichlet problem as a consequence we derive global integrability estimates for solutions with respect to lorentz and morrey norms

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

generalized scattering matrix analysis of a class of finite arrays of coupled antennas by using 3 d fem and spherical mode expansion a rigorous method for characterizing in terms of a multimode scattering matrix a finite array of antennas where each antenna can be described by means of spherical waves is presented in this paper the procedure provides the impedance coupling and radiating characteristics of the arrays and comprises two steps first the generalized scattering matrix gsm of each single antenna is numerically calculated over a wide band of frequencies for this purpose we use a previously developed methodology that combines the domain segmentation technique the three dimensional finite element method 3d fem spherical mode expansion and a reduced order model obtained using a symmetric matrix pade spl acute via lanczos sympvl algorithm second the overall gsm of the finite array is calculated starting from the gsm of each antenna and using rotation and translation of spherical waves a closed form expression for the overall gsm is given and different examples are shown in order to validate the proposed method

reverse khas minskii condition the aim of this paper is to present and discuss some equivalent characterizations of p parabolicity for complete riemannian manifolds in terms of existence of special exhaustion functions in particular khas minskii in ergodic properties of recurrent diffusion prossesses and stabilization of solution to the cauchy problem for parabolic equations theor prob appl 5 2 1960 proved that if there exists a 2 superharmonic function mathcal k defined outside a compact set on a complete riemannian manifold r such that lim x to infty mathcal k x infty then r is 2 parabolic and sario and nakai in classification theory of riemann surfaces springer berlin 1970 were able to improve this result by showing that r is 2 parabolic if and only if there exists an evans potential i e a 2 harmonic function e r setminus k to mathbb r with lim x to infty mathcal e x infty in this paper we will prove a reverse khas minskii condition valid for any p 1 and discuss the existence of evans potentials in the nonlinear case

holder estimates of p harmonic extension operators it is now a well known fact that for 1 p the p harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a 1 p poincare inequality are locally holder continuous in this note we provide a characterization of domains in such metric spaces for which p harmonic extensions of holder continuous boundary data are globally holder continuous we also provide a link between this regularity property of the domain and the uniform p fatness of the complement of the domain

p harmonic functions with boundary data having jump discontinuities and baernstein s problem for p harmonic functions on unweighted r 2 with 1 andlt p andlt infinity we show that if the boundary values f has a jump at an asymptotic corner point zo then the perron solution pf is asymptotically a b arg z z 0 o vertical bar z z 0 vertical bar we use this to obtain a positive answer to baernsteins problem on the equality of the p harmonic measure of a union g of open arcs on the boundary of the unit disc and the p harmonic measure of g over bar we also obtain various invariance results for functions with jumps and perturbations on small sets for p andgt 2 these results are new also for continuous functions finally we look at generalizations to r n and metric spaces

keller osserman a priori estimates and the harnack inequality for the evolution p laplace equation with singular absorption term abstract in this paper we study quasilinear equations of type u t d i v u p 2 u v x f u 0 p 2 u 0 despite the lack of comparison principle we prove a priori estimates of keller osserman type using these estimates we give a proof of the intrinsic harnack inequality

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

two theorems of n wiener for solutions of quasilinear elliptic equations relatively little is known about boundary behavior of solutions of quasilinear elliptic partial differential equations as compared to that of harmonic functions in this paper two results which in the harmonic case are due to n wiener are generalized to a nonlinear situation suppose that g is a bounded domain in r n we consider functions u g r which are free extremals of the variational integral

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

asymptotic uniqueness of ground states for a class of quasilinear schrodinger equations with h 1 supercritical exponent in this paper we obtain the asymptotic uniqueness of ground states for a class of quasilinear schrodinger equations with h1 supercritical exponent to this aim the uniqueness and the non degeneracy of positive solutions of the corresponding limit equation which is a zero mass problem play important roles thus we also show the uniqueness and non degeneracy result for a class of zero mass problems by applying ode techniques

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

sphericalization and p harmonic functions on unbounded domains in ahlfors regular spaces abstract we use sphericalization to study the dirichlet problem perron solutions and boundary regularity for p harmonic functions on unbounded sets in ahlfors regular metric spaces boundary regularity for the point at infinity is given special attention in particular we allow for several approach directions towards infinity and take into account the massiveness of their complements in 2005 llorente manfredi wu showed that the p harmonic measure on the upper half space r n n 2 is not subadditive on null sets when p 2 using their result and spherical inversion we create similar bounded examples in the unit ball b r n showing that the n harmonic measure is not subadditive on null sets when n 3 and neither are the p harmonic measures in b generated by certain weights depending on p 2 and n 2

harnack inequality and continuity of solutions to quasi linear degenerate parabolic equations with coefficients from kato type classes abstract for a general class of divergence type quasi linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear kato type classes we prove local boundedness and continuity of solutions and the intrinsic harnack inequality for positive solutions

the perron method for p harmonic functions in metric spaces we use the perron method to construct and study solutions of the dirichlet problem for p harmonic functions in proper metric measure spaces endowed with a doubling borel measure supporting a weak 1 q poincare inequality for some 1 q p the upper and lower perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p harmonic in the domain it is also shown that newtonian sobolev functions and continuous functions are resolutive i e that their upper and lower perron solutions coincide and that their perron solutions are invariant under perturbations of the function on a set of capacity zero we further study the problem of resolutivity and invariance under perturbations for semicontinuous functions we also characterize removable sets for bounded p super harmonic functions

boundary regularity for quasiminima in gg giaquinta giusti introduced the notion of quasiminima a concept which is very useful in unifying the c 0 regularity theory of elliptic equations in divergence form in particular their treatment provides regularity results for minima of nondifferentiable variational integrals they also show that weak solutions of nonlinear elliptic equations in divergence form under natural and general structural conditions are quasiminima as are solutions to a wide class of obstacle problems because of the great interplay between quasiminima and the holder theory of partial differential equations it is important to develop the subject fully indeed significant progress has already been made recently for example harnack inequalities have been established in dbt for sub and super quasiminima which run parallel to the results obtained by trudinger t for weak sub and supersolutions of quasilinear equations in divergence form in particular the results in dbt imply that quasiminima are holder continuous a result first established by giaquinta giusti gg

on the wiener criterion and quasilinear obstacle problems we study the wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators a a x vu vu 3 vu p in r local solutions are continuous at wiener points of the obstacle func tion if p n 1 the converse is also shown to be true if p n 1 then a characterization of the thinness of a set at a point is given in terms of a superharmonic functions 1 introduction in this paper we study the wiener criterion and variational

on p harmonic mappings in the plane abstract to every p harmonic vector field in the plane with p 2 there corresponds a quasilinear system of first order pdes which couples the complex gradients of the coordinate functions of the field the ellipticity of such system is proved a relation between planar quasiregular mappings and p harmonic fields is discussed the p harmonic conjugate problem is stated

a regularity classification of boundary points for p harmonic functions and quasiminimizers a regularity classification of boundary points for p harmonic funktions and quasiminimizers

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

the dirichlet problem for p harmonic functions on the topologist s comb in this paper we study the perron method for solving the p harmonic dirichlet problem on the topologist s comb for functions which are bounded and continuous at the accessible points we obtain invariance of the perron solutions under arbitrary perturbations on the set of inaccessible points we also obtain some results allowing for jumps and perturbations at a countable set of points

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

estimates at infinity for positive solutions to a class of p laplacian problems in rn abstract in this paper we consider a class of logistic type problems for the p laplacian in the whole space using minimization we prove existence of a positive solution and its behavior at infinity we also consider questions of uniqueness and sharp estimates at infinity

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

on p harmonic functions in the plane and their stream functions on considere l equation p harmonique p div p 2 0 1 p dans le plan on developpe un certain nombre d analogies avec la theorie de base des fonctions complexes dans c

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

quasilinear equations with source terms on carnot groups in this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of lane emden type with measure data on a carnot group g of arbitrary step the quasilinear part involves operators of the p laplacian type g p 1 p these results are based on new a priori estimates of solutions in terms of nonlinear potentials of th wolff s type as a consequence we characterize completely removable singularities and prove a liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub laplacian p 2 on the heisenberg group