A link between the shape of the austenite-martensite interface and the behaviour of the surface energy

Abstract

Let &#92;Omega&#92;subset&#92;mathbb{R}^{2} denote a bounded Lipschitz domain and consider some portion &#92;Gamma_{0} of &#92;partial&#92;Omega representing the austenite-twinned martensite interface which is not assumed to be a straight segment. We prove &#92;underset{u&#92;inmathcal{W}(&#92;Omega)}{&#92;mbox{inf}}&#92;int_{&#92;Omega}&#92;varphi(&#92;nabla u(x,y))dxdy=0 (*) for an elastic energy density &#92;varphi:&#92;mathbb{R}^{2}&#92;rightarrow[0,&#92;infty) such that &#92;varphi(0,&#92;pm1)=0. Here &#92;mathcal{W}(&#92;Omega) consists of all functions u from the Sobolev class W^{1,&#92;infty}(&#92;Omega) such that &#92;left|u_{y}&#92;right|=0 a.e. on &#92;Omega together with u=0 on &#92;Gamma_{0}. We will first show that for &#92;Gamma_{0} having a vertical tangent one cannot always expect a finite surface energy, i.e. in the above problem the condition u_{yy} is a Radon measure such that &#92;int_{&#92;Omega}&#92;left|u_{yy}(x,y)&#92;right|dxdy<+&#92;infty in general cannot be included. This generalizes a result of [W.] where &#92;Gamma_{0} is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary &#92;partial&#92;Omega, that is, one can even take &#92;Gamma_{0}=&#92;partial&#92;Omega. We also show that the existence or nonexistence of minimizers depends on the shape of the austenitetwinned martensite interface &#92;Gamma_{0}.