Non-subelliptic estimates for the tangential Cauchy–Riemann system

We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of $\mathbb{C}^{n}We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of . Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273–292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L ² estimates by H?rmander (Acta Math 113:89–152,1965) and Kohn (Trans AMS 181:273–292,1973) of Sect. 2 and the tangential -Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155–162,2000). The main technical result, Theorem 2.1, is a version for “perturbed” q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the -equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).To Prof. Giovanni Zacher in his 80th birthday.     

Single periodic Riemann problems for null-solutions to polynomially Cauchy–Riemann equations

In this paper, we focus on single periodic Riemann problems for a class of meta-analytic functions, i.e. null-solutions to polynomially Cauchy–Riemann equation. We first establish decomposition theorems for single periodic meta-analytic functions. Then, we give a series expansion of single periodic meta-analytic functions, and derive generalised Liouville theorems for them. Next, we introduce a definition of order for single periodic meta-analytic functions at infinity, and characterise their growth at infinity. Finally, applying the decomposition theorem for single periodic meta-analytic functions, we get explicit expressions of solutions and condition of solvability to Riemann problems for single periodic meta-analytic functions with a finite order at infinity.     

The Cauchy–Riemann equations on product domains

We establish the L ² theory for the Cauchy–Riemann equations on product domains provided that the Cauchy–Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.     

The range of the tangential Cauchy–Riemann system to a CR embedded manifold

We prove that every compact, pseudoconvex, orientable, CR manifold of , bounds a complex manifold in the C ^∞ sense. In particular, has closed range.     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.     

On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations

We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions in which uniqueness holds.     

Propagation of singularities for solutions of Hamilton–Jacobi equations

We consider the viscosity solution of the Cauchy problem for a class of Hamilton–Jacobi equations and we show that the points of the C¹$C^1$ singular support of such a function propagate along the generalized characteristics for all the times.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

Nonexistence of global solutions of the Cauchy problem for systems of klein–Gordon equations with positive initial energy

We study the Cauchy problem for systems of weakly coupled Klein–Gordon equations with dissipations. We prove a theorem on the nonexistence of global solutions with positive initial energy.     

The Cauchy–Riemann equations in the unit ball of l^{2}

We study the local exactness of the \(\overline{\partial }\) operator in the Hilbert space \(l^2\) for a particular class of \((0,1)\) -forms \(\omega \) of the type \(\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i\) , \(z = (z_i)\) in \(l^2\) . We suppose each function \(\omega ^i\) of class \(C^\infty \) in the closed unit ball of \(l^2\) , of the form \(\omega ^i(z) = \sum _k \omega ^i_k\left( z^k\right) \) , where \(\mathbf N = \bigcup I_k\) is a partition of \(\mathbf N\) , \((\) card \(I_k < +\infty )\) and \(z^k\) is the projection of \(z\) on \(\mathbf C^{I_k}\) . We establish sufficient conditions for exactness of \(\omega \) related to the expansion in Fourier series of the functions \(\omega ^i_k\) .     

New existence of multi-spike solutions for the fractional Schr?dinger equations

We consider the following fractional Schr¨odinger equation:(-?)^su + V(y)u = u^p, u > 0 in R^N,(0.1) where s ∈(0, 1), 1 < p 相似文献   

On soliton solutions for Boussinesq–Burgers equations

The periodic wave solutions for Boussinesq–Burgers equations are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

Minimax solutions of Hamilton–Jacobi functional equations for neutral-type systems

A functional Hamilton–Jacobi equation with covariant derivatives which corresponds to neutral-type dynamical systems is obtained. The definition of a minimax solution of this equation is given. Conditions under which such a solution exists and is unique and well defined are found.     

On L^{r} hypoellipticity of solutions with compact support of the Cauchy–Riemann equation

In one complex variable, the existence of a compactly supported solution to the Cauchy–Riemann equation is related to the vanishing of certain integrals of the data; trying to generalize this approach, we find an explicit construction, via convolution, for a compactly supported solution in \(\mathbb C ^n\) , which allows us to estimate the \(L^p\) norm of the solution. We also investigate the possible generalizations of this method to domains of the form \(P\setminus Z\) , where \(P\) is a polydisc and \(Z\) is the zero locus of some holomorphic function.     

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.