regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

the variable coefficient thin obstacle problem carleman inequalities abstract in this article we present a new strategy of addressing the variable coefficient thin obstacle problem our approach is based on a variable coefficient carleman estimate this yields semi continuity of the vanishing order lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow up procedure moreover the carleman estimate implies the existence of homogeneous blow up limits along certain sequences and ultimately leads to an almost optimal regularity statement as it is a very robust tool it allows us to consider the problem in the setting of sobolev metrics i e working on the upper half ball b 1 r n 1 the coefficients are only w 1 p regular for some p n 1 these results provide the basis for our further analysis of the free boundary the optimal regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow up article 21 in the framework of w 1 p p 2 n 1 regular coefficients and w 2 p p 2 n 1 regular non zero obstacles

the structure of the free boundary for lower dimensional obstacle problems we study the regularity of the free surface in boundary obstacle problems we show that near a non degenerate point the free boundary is a c 1 a n 2 dimensional surface in r n 1

pure odd order oscillators with constant excitation abstract in this paper the excited vibrations of a truly nonlinear oscillator are analyzed the excitation is assumed to be constant and the nonlinearity is pure without a linear term the mathematical model is a second order nonhomogeneous differential equation with strong nonlinear term using the first integral the exact value of period of vibration i e angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained the closed form solution has the form of gamma function the period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term for the case of pure odd order oscillators the approximate solution of differential equation is obtained in the form of trigonometric function the solution is based on the exact value of period of vibration for the case when additional small perturbation of the pure oscillator acts the so called cveticanin s averaging method for a truly nonlinear oscillator is applied two special cases are considered one when the additional term is a function of distance and the second when damping acts to prove the correctness of the method the obtained results are compared with those for the linear oscillator example of pure cubic oscillator with constant excitation and linear damping is widely discussed comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement the investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems for example such mechanical systems are seats in vehicles supports for machines cutting machines with periodical motion of the cutting tools presses etc the examples can be find in electronics electromechanical devices like micro actuators and micro oscillators in music instruments hammers in piano in human voice producing folds voice cords etc

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

optimal regularity for the signorini problem and its free boundary we will show optimal regularity for minimizers of the signorini problem for the lame system in particular if mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n minimizes begin aligned j mathbf u int b 1 nabla mathbf u nabla bot mathbf u 2 lambda hbox div mathbf u 2 end aligned in the convex set begin aligned k left mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n u n ge 0 quad hbox on quad pi right left mathbf u f in c infty partial b 1 quad hbox on quad partial b 1 right end aligned where lambda ge 0 say then mathbf u in c 1 1 2 b 1 2 moreover the free boundary given by begin aligned gamma mathbf u partial x u n x 0 x n 0 cap b 1 end aligned will be a c 1 alpha graph close to points where mathbf u is not degenerate similar results have been know before for scalar partial differential equations see for instance 5 6 the novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations

new monotonicity formulas and the optimal regularity in the signorini problem with variable coefficients abstract we study the interior signorini or lower dimensional obstacle problem for a uniformly elliptic divergence form operator l div a x with lipschitz continuous coefficients our main result states that similarly to what happens when l the variational solution has the optimal interior regularity c loc 1 1 2 m when m is a codimension one flat manifold which supports the obstacle we achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated almgren s frequency functional

the residue harmonic balance for fractional order van der pol like oscillators when seeking a solution in series form the number of terms needed to satisfy some preset requirements is unknown in the beginning an iterative formulation is proposed so that when an approximation is available the number of effective terms can be doubled in one iteration by solving a set of linear equations this is a new extension of the newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series when fourier series is employed the method is called the residue harmonic balance in this paper the fractional order van der pol oscillator with fractional restoring and damping forces is considered the residue harmonic balance method is used for generating the higher order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator the highly accurate solutions to angular frequency and limit cycle of the fractional order van der pol equations are obtained analytically the results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives the influence of the fractional order on the geometry of the limit cycle is investigated for the first time

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

higher regularity of the free boundary in the elliptic signorini problem abstract in this paper we study the higher regularity of the free boundary for the elliptic signorini problem by using a partial hodograph legendre transformation we show that the regular part of the free boundary is real analytic the first complication in the study is the invertibility of the hodograph transform which is only c 0 1 2 which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points the second and main complication in the study is that the equation satisfied by the legendre transform is degenerate however the equation has a subelliptic structure and can be viewed as a perturbation of the baouendi grushin operator by using the l p theory available for that operator we can bootstrap the regularity of the legendre transform up to real analyticity which implies the real analyticity of the free boundary

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

construction of approximate analytical solutions to a new class of non linear oscillator equation the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire by assuming that harmonic balance will hold solutions are devised for a steady state limit cycle and or limit point motion a method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective fourier expansions the latter technique is actually a generalization of the method proposed by kryloff and bogoliuboff 1943

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

uniqueness and reconstruction for the fractional calderon problem with a single measurement abstract we show global uniqueness in the fractional calderon problem with a single measurement and with data on arbitrary possibly disjoint subsets of the exterior the previous work 10 considered the case of infinitely many measurements the method is again based on the strong uniqueness properties for the fractional equation this time combined with a unique continuation principle from sets of measure zero we also give a constructive procedure for determining an unknown potential from a single exterior measurement based on constructive versions of the unique continuation result that involve different regularization schemes