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Capillary drops: contact angle hysteresis and sticking drops. (English) Zbl 1168.76007
The authors study an interesting homogenization procedure for the capillary problem which describes a drop resting on a plane. The governing equation is (1) $$2H=\Gamma\rho/\sigma-\lambda$$, where $$H$$ denotes the mean curvature of the capillary interface $$\mathcal S$$, $$\Gamma$$ the gravitational potential, $$\rho$$ the local density and $$\sigma$$ the surface tension. The constant $$\lambda$$ is a Lagrange parameter which comes from the volume constraint of the liquid. The contact angle condition (boundary condition) is (2) $$\cos\gamma=\beta$$ on $$\partial\mathcal{S}$$, where $$\gamma$$ is the angle between the capillary surface and the supporting plane, and $$\beta$$ is the relative adhesion coefficient.
The paper concerns the case that (3) $$\beta=\beta(x/\varepsilon,y/\varepsilon)$$, $$\varepsilon>0$$ is small, with a $$\mathbb Z$$-periodic $$\beta(x,y)$$. Set $$\langle\beta\rangle=\int_{[0,1]^2}\beta(x,y)\,dx\,dy$$. In the case of zero gravity $$(\Gamma=0)$$, the main result is that there exists a solution of the capillary equation $$2H=-\lambda$$ on $$\mathcal S$$ which satisfies the boundary condition $$\cos\gamma\leq\langle\beta\rangle$$, provided the crucial assumption $$\min_y\max_x\beta(x,y)<\langle\beta\rangle$$ is satisfied. The proofs of the results are based on compactness results for viscosity solutions to (1)–(3). Viscosity solutions are minimizers of an energy functional $$J_\varepsilon$$ associated to (1)–(3). The existence of the minimizers follows by using appropriate barriers.

##### MSC:
 76B45 Capillarity (surface tension) for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 49Q05 Minimal surfaces and optimization
##### Keywords:
homogenization; compactness; energy functional; minimizers
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