371 research outputs found

    Local and nonlocal boundary conditions for Ī¼\mu-transmission and fractional elliptic pseudodifferential operators

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    A classical pseudodifferential operator PP on RnR^n satisfies the Ī¼\mu-transmission condition relative to a smooth open subset Ī©\Omega , when the symbol terms have a certain twisted parity on the normal to āˆ‚Ī©\partial\Omega . As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic PP in full scales of Sobolev spaces with a singularity dĪ¼āˆ’kd^{\mu -k}, d(x)=distā”(x,āˆ‚Ī©)d(x)=\operatorname{dist}(x,\partial\Omega). Examples include fractional Laplacians (āˆ’Ī”)a(-\Delta)^a and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on Rnāˆ–Ī©R^n\setminus \Omega reduce to problems supported on Ī©Ė‰\bar\Omega, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces Fp,qsF^s_{p,q} and Bp,qsB^s_{p,q}, including H\"older-Zygmund spaces Bāˆž,āˆžsB^s_{\infty ,\infty}. This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of (āˆ’Ī”)au=fāˆˆLāˆž(Ī©)(-\Delta)^au=f\in L_\infty (\Omega), uāˆˆdaCa(Ī©Ė‰)u\in d^aC^a(\bar\Omega) when 0<a<10<a<1, aā‰ 1/2a\ne 1/2 (in daCaāˆ’Ļµ(Ī©Ė‰)d^aC^{a-\epsilon}(\bar\Omega) when a=1/2a=1/2).Comment: Title slightly changed, 34 page

    Sufficient condition for existence of solutions for higher-order resonance boundary value problem with one-dimensional p-Laplacian

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    By using coincidence degree theory of Mawhin, existence results for some higher order resonance multipoint boundary value problems with one dimensional p-Laplacian operator are obtained

    A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

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    In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.&nbsp

    Twin iterative solutions for a fractional differential turbulent flow model

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    We investigate the existence of twin iterative solutions for a fractional p-Laplacian equation with nonlocal boundary conditions. Using the monotone iterative technique, we establish a new existence result on the maximal and minimal solutions under suitable nonlinear growth conditions. We also consider some interesting particular cases and give an example to illustrate our main results

    A survey on stationary problems, Green's functions and spectrum of Sturmā€“Liouville problem with nonlocal boundary conditions

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    In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

    Solvability of boundary value problems for singular quasi-Laplacian differential equations on the whole line

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    This paper is concerned with some integral type boundary value problems associated to second order singular differential equations with quasi-Laplacian on the whole line. The emphasis is put on the one-dimensionalĀ p-Laplacian termĀ Ā involving a nonnegative functionĀ ĻĀ that may be singular atĀ tĀ = 0 and such thatĀ . A Banach space and a nonlinear completely continuous operator are defined in this paper. By using the Schauder's fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established. An example is presented to illustrate the main theorem
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