Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions by Hamid El Bahja

“…We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time \(L^{\infty}\) bounds for the weak solutions.…”
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Estimates of the norm of the convolution operator in anisotropic Besov spaces with the dominated mixed derivative by K.K. Sadykova, N.T. Tleukhanova

“… In this paper, we investigate the boundedness of the norm of the convolution operator in Sobolev spaces with the dominated mixed derivative and anisotropic Nikolsky-Besov spaces. …”
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Acoustic scattering by fractal screens: mathematical formulations and wavenumber-explicit continuity and coercivity estimates by Chandler-Wilde, S, Hewett, D

“…Our analysis teases out the explicit wavenumber dependence of the continuity and coercivity constants of the boundary integral operators, viewed as mappings between fractional Sobolev spaces, this in part extending previous results of Ha-Duong [18, 19]. …”

On the solution of evolution p(.)-Bilaplace equation with variable by Abderrazek Chaoui, Manal Djaghout

“…The well-posedness at each time step of the problem in suitable Lebesgue Sobolev spaces with variable exponent with the help of nonlinear monotone operators theory is investigated. …”
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Variational approach for a Steklov problem involving nonstandard growth conditions by Zehra Yucedag

“…The aim of this paper is to study the multiplicity of solutions for a nonlocal p(x)-Kirchhoff type problem with Steklov boundary value in variable exponent Sobolev spaces. We prove the existence of at least three solutions and a nontrivial weak solution of the problem, using the Ricceri's three critical points theorem together with Mountain Pass theorem.…”
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Fredholm property of regular hypoelliptic operators on the scales of multianisotropic spaces by Tumanyan Ani

“…We establish necessary and sufficient conditions for a priori estimates for differential operators acting in multianisotropic Sobolev spaces in Rn. Fredholm criteria and index invariance are obtained for a wide class of regular hypoelliptic operators on the special scales of multianisotropic weighted spaces.…”
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Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity by Hocine Ayadi, Fares Mokhtari

“…In this article, we prove the existence and the regularity of distributional solutions for a class of nonlinear anisotropic elliptic equations with $p_i(x)$ growth conditions, degenerate coercivity and $L^{m(\cdot)}$ data, with $m(\cdot)$ being small, in appropriate Lebesgue-Sobolev spaces with variable exponents. The obtained results extend some existing ones [8,10].…”
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Variational inequalities with the duality operator in Banach spaces by In-Sook Kim

“…As applications, the problem is discussed in the Lebesgue spaces L p and the Sobolev spaces W1,2 .…”
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Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions by Cholmin Sin, Sin-Il Ri

“…To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.…”
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Elliptic and Parabolic Equations with Involution and Degeneration at Higher Derivatives by Aleksandr I. Kozhanov, Oksana I. Bzheumikhova

“…We study the solvability in Sobolev spaces of boundary value problems for elliptic and parabolic equations with variable coefficients in the presence of an involution (involutive deviation) at higher derivatives, both in the nondegenerate and degenerate cases. …”
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Weak lower semicontinuity of variational functionals with variable growth by Yongqiang Fu

“…<p>Abstract</p> <p>In this paper, we establish the weak lower semicontinuity of variational functionals with variable growth in variable exponent Sobolev spaces. The weak lower semicontinuity is interesting by itself and can be applied to obtain the existence of an equilibrium solution in nonlinear elasticity.…”
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On weighted second order Adams inequalities with Navier boundary conditions by Federica Sani

“…We obtain some sharp weighted version of Adams' inequality on second order Sobolev spaces with Navier boundary conditions. The weights that we consider determine a supercritical exponential growth, except in the origin, and the corresponding inequalities hold on radial functions only. …”
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Calculus of variations in the mixed smoothness setting by Prosinski, A

“…<p>The present work constitutes a first step towards establishing a systematic framework for treating variational problems posed in Sobolev spaces of mixed smoothness. The crucial difference that separates the following from the existing body of work is that the functionals we consider here depend on the argument function through a mixture of derivatives of different orders in different directions.…”

Existence results for a nonlinear elliptic transmission problem of $p(x)$-Kirchhoff type by Eugenio Cabanillas Lapa, Felix Leon Barboza, Juan Benito Bernui Barros, Benigno Godoy Torres

“…We get our results by means of the monotone operator theory and the $(S_{+})$ mapping theory; the weak formulation takes place in suitable variable exponent Sobolev spaces.…”
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The well-posedness for semilinear time fractional wave equations on R^N by Yong Zhou, Jia Wei He, Ahmed Alsaedi, Bashir Ahmad

“…Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $ L^2 $-solutions for linear and semilinear problems. …”
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Matrix Hölder's inequality and divergence formulation of optimal transport of vector measures by Ciosmak, KJ

“…We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.…”

Nonlinear elliptic problems involving the generalized p(u)-Laplacian operator with Fourier boundary condition by Said Ait Temghart, Chakir Allalou, Khalid Hilal

“…We get the results by assuming the right-hand side function f to be an integrable function, and by using the regularization approach combined with the theory of Sobolev spaces with variable exponents. …”
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On the Transmission Problem for Second-Order Quasilinear Elliptic Equations in Divergence Form by M.M. Karchevsky, R.R. Shagidullin

“…A statement of the transmission problem for quasilinear elliptic equations in divergence form in bounded composed domains in terms of the strengthened Sobolev spaces is proposed. Some generalized sufficient solvability conditions for the Dirichlet boundary value problem are obtained. …”
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On Some Properties of Integral-Type Operator in Weighted Herz Spaces with Variable Exponent Lebesgue Spaces by Lütfi Akın

“…For the last quarter century a considerable number of research has been carried out on the study of Herz spaces, variable exponent Lebesgue  spaces and Sobolev spaces. This studies also have played an important role in  problems of elasticity, fluid dynamics, calculus of variations. …”
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RATIONALE OF DIFFERENTIAL SPLITTING SCHEMES FOR EQUATIONS OF VISCOUS COMPLRESSIBLE FLUIDS MIXTURE MOVEMENT by N. A. Kucher, O. D. Kiseleva

“…Prove the convergence at the scale of Sobolev spaces in the proposed scheme of splitting. The results can be used as the basis on the construction of mathematical analysis and the corresponding finite-difference splitting scheme. …”
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