MULTI-TERM TIME-FRACTIONAL DERIVATIVE HEAT EQUATION FOR ONE-DIMENSIONAL DUNKL OPERATOR by D. Serikbaev

“…In particular, we use the direct and inverse Dunkl transform to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The generalized solutions of this problem are studied.…”
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The topological degree methods for the fractional p(⋅)-Laplacian problems with discontinuous nonlinearities by Hasnae El Hammar, Chakir Allalou, Adil Abbassi, Abderrazak Kassidi

“…The appropriate functional framework for this problems is the fractional Sobolev space with variable exponent.…”
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Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations by A.V. Anufrieva, E.V. Rung, D.N. Tumakov

“…The concept of the generalized solution in the Sobolev space is formulated. Equivalence of the generalized and classical solutions is proven. …”
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Optimal quadrature formulas in the space W2​​(m,m−1) of periodic functions by Hayotov, A.R., Khayriev, U.N.

“…In addition, it is shown that the norm of the error functional for the optimal quadrature formula constructed in the space W2​​(m,m−1)is less than the value of the norm of the error functional for the optimal quadrature formula in the Sobolev space L2​​(m).…”
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Extremal functions for Morrey’s inequality in convex domains by Lindgren, Erik, Hynd, Ryan C

“…For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions is constant provided that Ω is convex. …”
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Distortion of Hausdorff measures under Orlicz-Sobolev maps by Cianchi, A, Korobkov, M, Kristensen, J

“…Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. …”

A stochastic McKean–Vlasov equation for absorbing diffusions on the half-line by Hambly, B, Ledger, S

“…Our techniques involve energy estimation in the dual of the first Sobolev space, which connects the regularity of solutions to their boundary behaviour, and tightness calculations in the Skorokhod M1 topology defined for distribution-valued processes, which exploits the monotonicity of the loss process L. …”

Stochastic PDEs for large portfolios with general mean-reverting volatility processes by Hambly, B, Kolliopoulos, N

“…The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. …”

Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian by Cheng Jun, Chen Peng, Zhang Limin

“…We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue-Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. …”
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Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power by Juan-Esteban Suarez Cardona, Michael Hecht

“…The formulations reflect classic Sobolev space theory for partial differential equations (PDEs) and their weak formulations. …”
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Observation problems posed for the Klein-Gordon equation by András Szijártó, Jenő Hegedűs

“…The essential conditions are the following: smoothness of $f, \ g$ as elements of a corresponding subspace $D^{s+i}(0,l)$ (introduced in [2]) of a Sobolev space $H^{s+i} (0,l)$, where $i=1,2$ depending on the type of the observation problem, and the representability of $t_2-t_1$ as a rational multiple of $\frac{2l}{a}$. …”
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Measure of the potential valleys of the supermembrane theory by Lyonell Boulton, María Pilar García del Moral, Álvaro Restuccia

“…This covers the important case of seven and eleven dimensional supermembrane theories, and implies the compact embedding of the Sobolev space H1,2(Ω) onto L2(Ω). The latter is a main step towards the confirmation of the existence and uniqueness of ground state solutions of the outer Dirichlet problem for the Hamiltonian of the SU(N) regularized D=11 supermembrane, and might eventually allow patching with the inner solutions.…”
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The global solution of anisotropic fourth-order Schrödinger equation by Hailing Su, Cuihua Guo

“…By using the Banach fixed point theorem, we obtain the existence, the uniqueness, the continuous dependence and the decay estimate of the solution on the initial value in anisotropic Sobolev spaces Hy→s1,ρHz→s2,r $H_{\vec{y}}^{s_{1},\rho } H_{\vec{z}} ^{s_{2},r}$.…”
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Global solutions to a nonlinear Fokker-Planck equation by Xingang Zhang, Zhe Liu, Ling Ding, Bo Tang

“…In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $ H^2_{x}L^2_v $ by a refined nonlinear energy method. …”
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On Landau-Kolmogorov type inequalities for charges and their applications by V.F. Babenko, V.V. Babenko, O.V. Kovalenko, N.V. Parfinovych

“…As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$ belongs to the Sobolev space $W^{1,\infty}$.…”
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The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces by Osvaldo Méndez

“…Abstract Given a Musielak–Orlicz function $$\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}$$ φ ( x , s ) : Ω × [ 0 , ∞ ) → R on a bounded regular domain $$\Omega \subset {\mathbb R}^n$$ Ω ⊂ R n and a continuous function $$M:[0,\infty )\rightarrow (0,\infty )$$ M : [ 0 , ∞ ) → ( 0 , ∞ ) , we show that the eigenvalue problem for the elliptic Kirchhoff’s equation $$-M\left( \int \limits _{\Omega }\varphi (x,|\nabla u(x)|)\textrm{d}x\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|} $$ - M ∫ Ω φ ( x , | ∇ u ( x ) | ) d x div ∂ φ ∂ s ( x , | ∇ u ( x ) | ) ∇ u ( x ) | ∇ u ( x ) | = λ ∂ φ ∂ s ( x , | u ( x ) | ) u ( x ) | u ( x ) | has infinitely many solutions in the Sobolev space $$W_0^{1,\varphi }(\Omega )$$ W 0 1 , φ ( Ω ) . …”
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Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters by Jun Lu, Lianpeng Shi, Chein-Shan Liu, C. S. Chen

“…The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. …”
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The Kuramoto–Sivashinsky equation. A Local Attractor Filled with Unstable Periodic Solutions by Anatoli N. Kulikov, Dmitri A. Kulikov

“…A spectrum of frequencies of the given family of periodic solutions fills the entire number line, and they are all unstable in a sense of Lyapunov definition in the metric of the phase space (space of initial conditions) of the corresponding initial boundary value problem. It is chosen the Sobolev space as the phase space. For the periodic solutions which fill the two-dimensional attractor, the asymptotic formulas are given. …”
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A shifted fractional-order Hahn functions Tau method for time-fractional PDE with nonsmooth solution by N. Mollahasani

“…Error and convergence analysis of the numerical method has been investigated in a Sobolev space. Finally, some numerical experiments are considered in the form of tables and figures to demonstrate the accuracy and capability of the proposed method.…”
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Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering by Angeliki Kaiafa, Vassilios Sevroglou

“…Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. …”
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