reducing the number of elements in the synthesis of shaped beam patterns by the forward backward matrix pencil method the matrix pencil method mpm has been used to reduce the number of elements in the linear antenna array with a pencil beam pattern this work extends the mpm based synthesis method to the synthesis of shaped beam patterns by using the forward backward matrix pencil method fbmpm the fbmpm based synthesis method places a necessary restriction on the poles which correspond to element positions and consequently obtains more accurate synthesis results particularly for the synthesis of asymmetric patterns numerical examples show the effectiveness and advantages of the proposed method in the reduction of the number of elements for shaped beam patterns

regularizing effect of absorption terms in singular problems abstract we prove existence of solutions to problems whose model is p u u q f u in u 0 in u 0 on where is an open bounded subset of r n n 2 p u is the p laplacian operator for 1 p n q 0 0 and f is a nonnegative function in l m for some m 1 in particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case 1 we also study uniqueness of these solutions as well as examples which show the optimality of the results finally we find local w 1 p solutions in case 1

fast pencil beam pattern synthesis of large unequally spaced antenna arrays reducing the computational cost of large array pattern synthesis is attractive in many applications in this paper a fast pencil beam pattern synthesis method for large nonuniform antenna arrays is proposed this method is based on an interpolation in a least square sense and iterative fast fourier transform fft i e interpolate the nonuniform distributed elements into uniform virtual elements and then apply fft to synthesis the uniform array thanks to the efficiency of fft the proposed method is much faster and can handle much larger arrays than the existing methods to guarantee the interpolation accuracy the choice of the interpolation parameters is discussed both linear and planar nonuniform array examples are shown to validate the advantages of the proposed method

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

a hybridizable discontinuous galerkin method for the p laplacian we propose the first hybridizable discontinuous galerkin method for the p laplacian equation when using polynomials of degree k geq 0 for the approximation spaces of u boldsymbol nabla u and boldsymbol nabla u p 2 boldsymbol nabla u the method exhibits optimal k 1 order of convergence for all variables in l 1 and l p norms in our numerical experiments for k geq 1 an elementwise computation allows us to obtain a new approximation u h that converges to u with order k 2 we rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms the unknown of the first problem is the approximation of u on the skeleton of the mesh but requires solving nonlinear local problems the second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems we present numerical results displaying the convergence properties of the methods demonstrating the utility of

anzellotti s pairing theory and the gauss green theorem abstract in this paper we obtain a very general gauss green formula for weakly differentiable functions and sets of finite perimeter this result is obtained by revisiting anzellotti s pairing theory and by characterizing the measure pairing a d u when a is a bounded divergence measure vector field and u is a bounded function of bounded variation

numerical solution of boundary value problems for the eikonal equation in an anisotropic medium abstract a dirichlet problem is considered for the eikonal equation in an anisotropic medium the nonlinear boundary value problem bvp formulated in the present work is the limit of the diffusion reaction problem with a diffusion parameter tending to zero to solve numerically the singularly perturbed diffusion reaction problem monotone approximations are employed numerical examples are presented for a two dimensional bvp for the eikonal equation in an anisotropic medium the standard lagrangian finite element approximation in space is used in computations

absolute continuity of the best sobolev constant abstract let q inf u l p p u l q p u w 0 1 p 0 be the best sobolev constant of the immersion w 0 1 p l q where is a bounded and smooth domain of r n 1 p n and 1 q p n p n p we prove that the function q 1 p q is absolutely continuous

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

on the solutions to 1 laplacian equation with l1 data in the present paper we study the behaviour as p goes to 1 of the renormalized solutions to the problems 0 1 div up p 2 up fin up 0on where p 1 is a bounded open set of rn n 2 with lipschitz boundary and f belongs to l1 we prove that these renormalized solutions pointwise converge up to subsequences to a function u with a suitable definition of solution we also prove that u is a solution to a limit problem moreover we analyze the situation occurring when more regular data f are considered

generalized analytical technique for the synthesis of unequally spaced arrays with linear planar cylindrical or spherical geometry an effective method for optimizing the performance of a fixed current distribution uniformly spaced antenna array has been to adjust its element positions to provide performance improvement in comparison with the default uniform structure this approach yields performance improvements such as smaller sidelobe levels or beamwidth values additionally it provides practical advantages such as reductions in size weight and number of antenna elements the objective of this paper is to describe a unified mathematical approach to nonlinear optimization of multidimensional array geometries the approach utilizes a class of limiting properties of sinusoidal bessel or legendre functions that are dictated by the array geometry addressed the efficacy of the method is demonstrated by its generalized application to synthesis of rectangular cylindrical and spherical arrays the unified mathematical approach presented below is a synthesis technique founded on the mathematical transformation of the desired field pattern followed by the application of limiting forms of the transformation and resulting in the development of a closed form expression for the element positions the method offers the following advantages over previous techniques such as direct nonlinear optimization or genetic algorithms first it is not an iterative searching algorithm and provides element spacing values directly in a single run of the algorithm thereby saving valuable cpu time and memory storage second it permits the array designer to place practical constraints on the array geometry e g the minimum maximum spacing between adjacent elements

overdetermined problems with possibly degenerate ellipticity a geometric approach given an open bounded connected subset of n we consider the overdetermined boundary value problem obtained by adding both zero dirichlet and constant neumann boundary data to the elliptic equation div a u u 1 in we prove that if this problem admits a solution in a suitable weak sense then is a ball this is obtained under fairly general assumptions on and a in particular a may be degenerate and no growth condition is required our method of proof is quite simple it relies on a maximum principle for a suitable p function combined with some geometric arguments involving the mean curvature of

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

asymmetric affine lp sobolev inequalities abstract a new sharp affine l p sobolev inequality for functions on r n is established this inequality strengthens and implies the previously known affine l p sobolev inequality which in turn is stronger than the classical l p sobolev inequality

classification of bifurcation diagrams of a p laplacian dirichlet problem with examples abstract we study bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 x 1 u 1 u 1 0 where p y y p 2 y p u is the one dimensional p laplacian and p 1 and 0 are two bifurcation parameters assume that f u g u h u where g h c 0 c 2 0 satisfy hypotheses h1 h5 presented herein for different values p with 1 p 2 and with p 2 we give a classification of totally six different bifurcation diagrams we prove that on the u plane each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right hence we are able to determine the exact multiplicity of positive solutions in addition for 1 p 2 and for p 2 we give interesting examples f u k u p 1 u q u r satisfying r q p 1 and k 0 and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to

on the shape of the nonnegative solutions to a singularly perturbed quasilinear dirichlet problem abstract we study the shape of the nontrivial nonnegative solutions for a singularly perturbed quasilinear dirichlet problem with non lipschitz nonlinearity we find that there exists at least one nontrivial nonnegative solution of the problem which consists of some bubbles the profile of each bubble is discussed

the santalo regions of a convex body motivated by the blaschke santalo inequality we define for a convex body k in r and for t r the santalo regions s k t of k we investigate properties of these sets and relate them to a concept of affine differential geometry the affine surface area of k let k be a convex body in r for x int k the interior of k let k be the polar body of k with respect to x it is well known that there exists a unique x0 int k such that the product of the volumes k k0 is minimal see for instance sch this unique x0 is called the santalo point of k moreover the blaschke santalo inequality says that k k0 v n where vn denotes the volume of the n dimensional euclidean unit ball b 0 1 with equality if and only if k is an ellipsoid for t r we consider here the sets s k t x k k k v2 n t following e lutwak we call s k t a santalo region of k observe that it follows from the blaschke santalo inequality that the santalopoint x0 s k 1 and that s k 1 x0 if and only if k is an ellipsoid thus s k t has non empty interior for some t 1 if and only if k is not an ellipsoid in the first part of this paper we show some properties of s k t and give estimates on the size of s k t this question was asked by e lutwak the paper was written while both authors stayed at msri supported by a grant from the national science foundation msc classification 52

a neumann problem for the p x laplacian with p 1 in a subdomain abstract in this paper we study a neumann problem with non homogeneous boundary conditions for the p x laplacian in particular we assume that p is a step function defined in a domain and equals to 1 in a subdomain 1 and 2 in its complementary 2 by considering a suitable sequence p k of variable exponents such that p k p and replacing p with p k in the original problem we prove the existence of a solution u k for each of those intermediate ones we also show that under a hypothesis concerning the boundary data g the limit of the sequence u k is a function u which belongs to the space of functions of bounded variation and is a solution to the original p problem

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

synthesis of unequally spaced antenna arrays by using differential evolution synthesis of unequally spaced linear antenna arrays is considered in this paper a recently developed new differential evolution algorithm is applied to solve the problem both position only and position phase synthesis have been studied effect of angle resolution has also been investigated synthesis results show that our algorithm is able to obtain better synthesis result reliably and efficiently

near chebyshev pattern for nonuniformly spaced arrays using zeros matching method an analytic method is presented to design nonuniformly spaced uniform amplitude antenna arrays nsuaas to have near chebyshev radiation pattern in this method called zeros matching method zmm the zeros of array factors of nsuaas are matched with the zeros of the exact chebyshev pattern some examples are brought to verify the effectiveness of the zmm approach for both linear and planar nsuaas

relaxation of hamilton jacobi equations we study the relaxation of hamilton jacobi equations the relaxation in our terminology is the following phenomenon the pointwise supremum over a certain collection of subsolutions in the almost everywhere sense of a hamilton jacobi equation yields a viscosity solution of the convexified hamilton jacobi equation this phenomenon has recently been observed in 13 in eikonal equations we show in this paper that this relaxation is a common phenomenon for a wide range of hamilton jacobi equations

spike layered solutions with compact support to some singularly perturbed quasilinear elliptic problems in general smooth domains in this paper we study the structure of nonnegative nontrivial solutions of the following problem e pu f u in u 0 in u 0 on as e 0 where pu div du p 2du with p 1 e 0 is a small parameter and is a bounded smooth domain in rn n 1 f is a class of logistic type nonlinearities satisfying f 0 f z1 f z2 0 with 0 0 in z1 z2 and limu 0 f u up 1 by virtue of the sub and supersolution method we prove that there are many nonnegative nontrivial solutions and they are spike layered solutions moreover the measure of each spike layer is estimated as e 0

solutions of the cheeger problem via torsion functions abstract the cheeger problem for a bounded domain r n n 1 consists in minimizing the quotients e e among all smooth subdomains e and the cheeger constant h is the minimum of these quotients let p c 1 be the p torsion function that is the solution of torsional creep problem p p 1 in p 0 on where p u div u p 2 u is the p laplacian operator p 1 the paper emphasizes the connection between these problems we prove that lim p 1 p l 1 p h lim p 1 p l 1 1 p moreover we deduce the relation lim p 1 p l 1 c n lim p 1 p l where c n is a constant depending only of n and h explicitely given in the paper an eigenfunction u bv l of the dirichlet 1 laplacian is obtained as the strong l 1 limit as p 1 of a subsequence of the family p p l 1 p 1 almost all t level sets e t of u are cheeger sets and our estimates of u on the cheeger set e 0 yield b 1 h b 1 n e 0 h n where b 1 is the unit ball in r n for convex we obtain u e 0 1 e 0

the element free galerkin method for the nonlinear p laplacian equation abstract the element free galerkin efg method is developed in this paper for solving the nonlinear p laplacian equation the moving least squares approximation is used to generate meshless shape functions the penalty approach is adopted to enforce the dirichlet boundary condition the galerkin weak form is employed to obtain the system of discrete equations and two iterative procedures are developed to deal with the strong nonlinearity then the computational formulas of the efg method for the p laplacian equation are established numerical results are finally given to verify the convergence and high computational precision of the method

the convective eigenvalues of the one dimensional p laplacian as p 1 abstract this paper studies the limit behavior as p 1 of the eigenvalue problem u x p 2 u x x c u x p 2 u x u p 2 u 0 x 1 u 0 u 1 0 we point out that explicit expressions for both the eigenvalues n and associated eigenfunctions are not available see 16 in spite of this hindrance we obtain the precise values of the limits lim p 1 n in addition a complete description of the limit profiles of the eigenfunctions is accomplished moreover the formal limit problem as p 1 is also addressed the results extend known features for the special case c 0 6 28

anisotropic p q laplacian equations when p goes to 1 abstract in this paper we prove a stability result for an anisotropic elliptic problem more precisely we consider the dirichlet problem for an anisotropic equation which is as the p laplacian equation with respect to a group of variables and as the q laplacian equation with respect to the other variables 1 p q with datum f belonging to a suitable lebesgue space for this problem we study the behaviour of the solutions as p goes to 1 showing that they converge to a function u which is almost everywhere finite regardless of the size of the datum f moreover we prove that this u is the unique solution of a limit problem having the 1 laplacian operator with respect to the first group of variables furthermore the regularity of the solutions to the limit problem is studied and explicit examples are shown

an extension of the pairing theory between divergence measure fields and bv functions abstract in this paper we introduce a nonlinear version of the notion of anzellotti s pairing between divergence measure vector fields and functions of bounded variation motivated by possible applications to evolutionary quasilinear problems as a consequence of our analysis we prove a generalized gauss green formula

sidelobe reduction in array pattern synthesis using genetic algorithm a simple and flexible genetic algorithm ga for pattern synthesis of antenna array with arbitrary geometric configuration is presented unlike conventional ga using binary coding and binary crossover this approach directly represents the array excitation weighting vectors as complex number chromosomes and uses decimal linear crossover without a crossover site compared with conventional gas this approach has a few advantages giving a clearer and simpler representation of the problem simplifying chromosome construction and totally avoiding binary encoding and decoding so as to simplify software programming and to reduce cpu time this method also allows us to impose constraints on phases and magnitudes of complex excitation coefficients for preferable implementation in practice using digital phase shifters and digital attenuators successful applications show that the approach can be used as a general tool for pattern synthesis of arbitrary arrays

asymmetric blaschke santalo functional inequalities abstract in this work we establish functional asymmetric versions of the celebrated blaschke santalo inequality as consequences of these inequalities we recover their geometric counter parts with equality cases as well as another inequality with strong probabilistic flavor that was firstly obtained by lutwak yang and zhang we present a brief study on an l p functional analogue to the center of mass that is necessary for our arguments and that might be of independent interest

uniqueness of the cheeger set of a convex body abstract we prove that if c r n is an open bounded convex set then there is only one cheeger set inside c and it is convex a cheeger set of c is a set which minimizes the ratio perimeter over volume among all subsets of c

a hk mortar spectral element method for the p laplacian equation abstract we present a constrained vertex variant of the mortar spectral element method to solve the p laplacian equation to show reliability of the method first we investigate convergence rate of h version and k version for a sufficiently smooth solution then for solutions with limited regularity we use a h k strategy in which the size of elements is geometrically reduced towards the singularity while the polynomial degree is linearly reduced in fact we numerically study convergence rate of the h k mortar spectral element method in the l 2 norm and h 1 norm using geometrical meshes with elements of non uniform polynomial degree in this regard we consider several benchmarks with various choices of the parameter p as well as geometric grading factors to deal with possible degeneracy in the p laplacian operator we add artificial diffusion to the original equation and then study the effects of this extra diffusion on convergence rate to find the solution of highly nonlinear system arising from discretization we use an optimization technique based on the trust region method in which an analytical jacobian matrix is utilized

peak sidelobe level reduction with a hybrid approach based on gas and difference sets this paper presents an approach for the optimization of the beam pattern produced by massively thinned arrays the method which combines the most attractive features of a genetic algorithm and those of a combinatorial technique namely the difference sets method is aimed at synthesizing massively thinned antenna arrays in order to suitably reduce the peak sidelobe level selected numerical results are presented in order to assess the effectiveness and reliability of the proposed approach

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

lyz ellipsoid and petty projection body for log concave functions abstract the aims of this paper are to develop the lyz ellipsoid and petty projection body for log concave functions which correspond to the lyz ellipsoid and petty projection body for convex bodies when restricted to the subclass of characteristic functions moreover the continuous sl n contravariant valuation on a subclass of log concave functions is classified

decay rate estimates for the quasi linear timoshenko system with nonlinear control and damping terms in this work we consider a class of quasi linear timoshenko systems with nonlinear control and damping terms we investigate the stabilization of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on control and damping terms

weak solutions of quasilinear problems with nonlinear boundary condition the growing attention for the study of the p laplacian operator p in the last few decades is motivated by the fact that it arises in various applications for instance in fluid mechanics the shear stress and the velocity gradient pu of certain uids obey a relation of the form x a x pu x where pu u p 2 u here p 1 is an arbitrary real number and the case p 2 respectively pi 2 p 2 corresponds to a newtonian respectively pseudoplastic dilatant uid the resulting equations of motion then involve div a pu which reduces to a pu a div pu provided that a is constant the p laplacian also appears in the study of torsional creep elastic for p 2 plastic as p see 7 ow through porous media p 2 see 12 or glacial sliding p 1 3 see 9 let rn be an unbounded domain with possible noncompact smooth boundary and n is the unit outward normal on we consider the nonlinear elliptic boundary value problem div a x u p 2 u 1 x 1 u p 2u 1 x 2 u q 2u in

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

on a family of torsional creep problems involving rapidly growing operators in divergence form let n n 2 be a bounded domain with smooth boundary and p n be a sequence of real numbers converging to as n for each integer n 1 we define the function varphi n t p n vert t vert p n 2 te vert t vert p n for all t and we prove the existence of a unique nonnegative variational solution for the problem div n u x u x u x n 1 when x subject to the homogeneous dirichlet boundary condition next we establish the uniform convergence in of the sequence of solutions for the above family of equations to the distance function to the boundary of our result complements the earlier developments on the topic obtained by payne and philippin 26 kawohl 21 bhattacharya dibenedetto and manfredi 2 perez llanos and rossi 27 and bocea and mih ilescu 4

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

elliptic equations involving the 1 laplacian and a subcritical source term abstract in this paper we deal with a dirichlet problem for an elliptic equation involving the 1 laplacian operator and a source term we prove that when the growth of the source is subcritical there exist two bounded nontrivial solutions to our problem moreover a pohozaev type identity is proved which holds even when the growth is supercritical we also show explicit examples of our results

some functional forms of blaschke santalo inequality we establish new functional versions of the blaschke santalo inequality on the volume product of a convex body which generalize to the non symmetric setting an inequality of ball isometric problems in l p and sections of convex sets phd dissertation cambridge 1986 and we give a simple proof of the case of equality as a corollary we get some inequalities for log concave functions and legendre transforms which extend the recent result of artstein et al mathematika 51 33 48 2004 with its equality case

computing the first eigenpair of the p laplacian in annuli abstract we propose a method for computing the first eigenpair of the dirichlet p laplacian p 1 in the annulus a b x r n a x b n 1 for each t a b we use an inverse iteration method to solve two radial eigenvalue problems one in the annulus a t with the corresponding eigenvalue t and boundary conditions u a 0 u t and the other in the annulus t b with the corresponding eigenvalue t and boundary conditions u t 0 u b next we adjust the parameter t using a matching procedure to make t coincide with t thereby obtaining the first eigenvalue p hence by a simple splicing argument we obtain the positive l normalized radial first eigenfunction u p the matching parameter is the maximum point of u p in order to apply this method we derive estimates for t and t and we prove that these functions are monotone and locally lipschitz continuous moreover we derive upper and lower estimates for the maximum point which we use in the matching procedure and we also present a direct proof that u p converges to the l normalized distance function to the boundary as p we also present some numerical results obtained using this method

pattern synthesis for arbitrary arrays using an adaptive array method this paper presents a new pattern synthesis algorithm for arbitrary arrays based on adaptive array theory with this algorithm the designer can efficiently control both mainlobe shaping and sidelobe levels the element weights optimize a weighted l sub 2 norm between desired and achieved patterns the values of the weighting function in the l sub 2 norm interpreted as imaginary jammers as in olen and compton s 1990 method are iterated to minimize exceedance of the desired sidelobe levels and minimize the absolute difference between desired and achieved mainlobe patterns the sidelobe control can be achieved by iteration only on sidelobe peaks in comparison to olen and compton s method the new algorithm provides a great improvement in mainlobe shaping control example simulations including both nonuniform linear and planar arrays are shown to illustrate the effectiveness of this algorithm

the sharp affine l 2 sobolev trace inequality and variants we establish a sharp affine l p sobolev trace inequality by using the l p busemann petty centroid inequality for p 2 our affine version is stronger than the famous sharp l 2 sobolev trace inequality proved independently by escobar and beckner our approach allows also to characterize all extremizers in this case for this new inequality no euclidean geometric structure is needed

minimizing within convex bodies using a convex hull method we present numerical methods to solve optimization problems on the space of convex functions or among convex bodies hence convexity is a constraint on the admissible objects whereas the functionals are not required to be convex to deal with this our method mixes geometrical and numerical algorithms we give several applications arising from classical problems in geometry and analysis alexandrov s problem of finding a convex body of prescribed surface function cheeger s problem of a subdomain minimizing the ratio surface area on volume newton s problem of the body of minimal resistance in particular for the latter application the minimizers are still unknown except in some particular classes we give approximate solutions better than the theoretical known ones hence demonstrating that the minimizers do not belong to these classes