Theorems on Isomorphisms for Some Parabolic Initial-Boundary-Value Problems in Hörmander Spaces: Limiting Case

Los, Valerii

doi:10.1007/s11253-016-1264-8

Cited by 11 publications

(5 citation statements)

References 23 publications

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“…The above-mentioned interpolation method proved to be useful in the theory of parabolic initial-boundary value problems as well. This was shown in papers [24,26,27] for some classes of parabolic problems. These papers deal with the generalized anisotropic Sobolev spaces H s,s/(2b);ϕ := B 2,µ , where…”

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confidence: 67%

Parabolic problems in generalized Sobolev spaces

Los¹,

Mikhailets²,

Murach³

2021

CPAA

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<p style='text-indent:20px;'>We consider a general inhomogeneous parabolic initial-boundary value problem for a <inline-formula><tex-math id="M1">\begin{document}$ 2b $\end{document}</tex-math></inline-formula>-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers <inline-formula><tex-math id="M2">\begin{document}$ s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ s/(2b) $\end{document}</tex-math></inline-formula> and with a function <inline-formula><tex-math id="M4">\begin{document}$ \varphi:[1,\infty)\to(0,\infty) $\end{document}</tex-math></inline-formula> that varies slowly at infinity. The function parameter <inline-formula><tex-math id="M5">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.</p>

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mentioning

confidence: 67%

Parabolic problems in generalized Sobolev spaces

Los¹,

Mikhailets²,

Murach³

2021

CPAA

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“…The above-mentioned interpolation method proved to be useful in the theory of parabolic initial-boundary value problems as well. This was shown in papers [28,30,[32][33][34] for some classes of parabolic problems. These papers deal with the generalized anisotropic Sobolev spaces H s,s/(2b);ϕ := B 2,µ , where…”

Section: Introductionmentioning

confidence: 86%

Parabolic problems in generalized Sobolev spaces

Los¹,

Mikhailets²,

Murach³

2019

Preprint

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We consider a general inhomogeneous parabolic initial-boundary value problem for a 2b-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers s and s/(2b) and with a function ϕ : [1, ∞) → (0, ∞) that varies slowly at infinity. The function parameter ϕ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.

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“…In recent years, other classes of function spaces are applied more and more actively to parabolic problems; namely, spaces with mixed norms, Triebel-Lizorkin spaces, weighted spaces, spaces of generalized smoothness (see, e.g., [5,6,11,15,23,33] and reference therein). Our paper continues the series of works [18][19][20][23][24][25] aimed to build a theory of parabolic initial-boundary value problems in function spaces of generalized anisotropic smoothness. This smoothness is given by a pair of real numbers and by a radial function which varies slowly at infinity and characterizes supplementary smoothness with respect to that given by the numbers.…”

Section: Introductionmentioning

confidence: 87%

Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

Dyachenko¹,

Los²

2021

Preprint

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We consider an inhomogeneous initial-boundary value problem for a Petrovskii parabolic system of second order PDEs. We prove that this problem induces isomorphisms between appropriate anisotropic generalized Sobolev spaces. The regularity of these spaces are given by a pair of real numbers and by a function parameter. The latter allows us to characterize the regularity of solutions to the problem more finely as compared with anisotropic Sobolev spaces.

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Theorems on Isomorphisms for Some Parabolic Initial-Boundary-Value Problems in Hörmander Spaces: Limiting Case

Cited by 11 publications

References 23 publications

Parabolic problems in generalized Sobolev spaces

Parabolic problems in generalized Sobolev spaces

Parabolic problems in generalized Sobolev spaces

Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

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