Variational Problems with Two Phases and Their Free Boundaries
نویسندگان
چکیده
The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered. Here q(x) ¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\]-\22) across the free boundary [u ¥= 0). Regularity and various properties are established for the minimizer u and for the free boundary. Introduction. In this paper we consider the problem of minimizing J{v) = f[\ Vv\2 + q2(x)X2(v)]dx, vGK,
منابع مشابه
Variational Iteration Method for Free Vibration Analysis of a Timoshenko Beam under Various Boundary Conditions
In this paper, a relatively new method, namely variational iteration method (VIM), is developed for free vibration analysis of a Timoshenko beam with different boundary conditions. In the VIM, an appropriate Lagrange multiplier is first chosen according to order of the governing differential equation of the boundary value problem, and then an iteration process is used till the desired accuracy ...
متن کاملDimensional Estimates for Singular Sets in Geometric Variational Problems with Free Boundaries
We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are Hn−3-negligible, where n is the ambient space dimension. In particular our results apply to capillarity type problems, and establish everywhere regularity in the three-dimensional case.
متن کاملRegularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law
Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young’s law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of ...
متن کاملAxial and Torsional Free Vibrations of Elastic Nano-Beams by Stress-Driven Two-Phase Elasticity
Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter...
متن کاملHartley Series Direct Method for Variational Problems
The computational method based on using the operational matrix of anorthogonal function for solving variational problems is computeroriented. In this approach, a truncated Hartley series together withthe operational matrix of integration and integration of the crossproduct of two cas vectors are used for finding the solution ofvariational problems. Two illustrative...
متن کامل