JOURNAL OF PARTIAL DIFFERENTIAL EQUATIONS J. Part. Diff. Eq., Vol. 33, No. 1, pp. 64-92 doi: 10.4208/jpde.v33.n1.5 March 2020 Explicit H 1 -Estimate for the Solution of the Lam ´ e System with Mixed Boundary Conditions AIT-AKLI Djamel ∗ and MERAKEB Abdelkader L2CSP, Mouloud Mammeri University Tizi-Ouzou, 15000, Algeria. Received 20 May 2019; Accepted 27 February 2020 Abstract. In this paper we consider the Lam´ e system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type. An explicit L 2 norm estimate for the gradient of the solution of this problem is established. This leads to an explicit bound of the H 1 norm of this solution. Note that the obtained upper-bound is not optimal. AMS Subject Classifications: 35J57, 74B05 Chinese Library Classifications: O175.27 Key Words: Lam´ e system; Korn’s inequality; Poincare’s inequality; inequality of trace; explicit estimates. 1 Introduction Let Ω be a bounded open connected subset of R 2 . The static equilibrium of a deformable structure occupying Ω is governed by the Lam´ e linear elasto-static system, see [1]. In this paper, we restrict the study to a convex domain Ω whose boundary has a polygonal shape that posses m+1 edges with m ≥ 2. We denote Γ = ∪ m i=0 Γ i its boundary and d(Ω) its diameter. Moreover, we assume that all the edges Γ i have strictly positive measure. The system under consideration is given by      Lu = f a.e in Ω, σ(u) · −→ n i = g i on (Γ − Γ 0 ) ∩ Γ i ,1 ≤ i ≤ m, u = 0 on Γ 0 . (1.1) ∗ Corresponding author. Email addresses: djamel.aitakli@ummto.dz (D. Ait-akli), kader.merakeb@ummto.dz (A. Merakeb) http://www.global-sci.org/jpde/ 64