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A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón–Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón–Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical (- Δ p) 1 -Laplace operator with 1 < p < ∞ , is measurable in one variable and has a small BMO assumption for the other variables, while the singular kernel associated with the nonlocal (- Δ q) s -Laplace operator with 0 < s < 1 < q < ∞ is merely measurable, and that the boundary of the domain is sufficiently flat in Reifenberg sense, we prove that the gradient of the solution is as integrable as that of the associated non-homogeneous term in the divergence form. Our regularity theory reported here could be a nice addition for higher integrability of the gradient to the Hölder continuity of the gradient in De Filippis and Mingione (Math Ann https://doi.org/10.1007/s00208-022-02512-7, 2022) regarding mixed local and nonlocal problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized q-step scheme with increasing convergence order 2q + 2 is proposed and analyzed. Novelty of the q-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation. [ABSTRACT FROM AUTHOR]

Actuation devices made of smart polymers typically show various instabilities, which can adversely affect their performance and lead to device failure. In general, smart polymers exhibit wrinkling instability when subjected to an electric or magnetic field. At the same time, wrinkles can be used constructively in certain applications demanding a controlled alternation of the surface morphology. Critical factors influencing thin films' pull-in and wrinkling instabilities are discovered concerning the anisotropic taut domains with an applied electro-magneto-mechanical field control. A continuum mechanics-based electro-magneto-mechanical model is developed for predicting the thresholds on the taut domains in the plane of principal stretches. Also, the concept of natural width under simple tension is implemented to derive the coupled nonlinear equation that evaluates the associated taut domains. The findings of the model solution indicate that the extent of taut domains can be controlled by modifying the level and the principal direction of the transverse isotropy. Additionally, the taut domain for a particular level of applied electromagnetic field increases with an increase in the anisotropy parameter, while it depleted with an increase in the fiber orientations from 0 ° to 90 ° for an applied level of electromagnetic loading. • A taut domains model of electro-magneto-active smart thin membrane is developed. • The concept of natural width in tension is applied to derive the governing equation. • The proposed model is validated with an energy-based model of dielectric membrane. • The effect of anisotropy and fiber orientations on the taut domains is investigated. [ABSTRACT FROM AUTHOR]

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric. [ABSTRACT FROM AUTHOR]