By William Martin Baker

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Habilitation ` a diriger des recherches, Paris VI. BERNSTEIN AND DE GIORGI TYPE PROBLEMS [Far03] [Far07] [FCS80] [GG98] [GT01] [LL97] [Mod85] [MP78] [Sav03] [Ser94] [Sim07] [SZ98a] [SZ98b] [Tol84] [Uhl77] [VSS06] 47 Alberto Farina. One-dimensional symmetry for solutions of quasilinear equations in R2 . Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6(3):685–692, 2003. Alberto Farina. Liouville-type theorems for elliptic problems. In M. Chipot, editor, Handbook of Differential Equations: Stationary Partial Differential Equations.

Then, given φ ∈ C0∞ (R3 ), and ǫ > 0, let η ∈ C ∞ (R) be such that η(r) = η(−r), η(r) = 0 if |r| ≤ 1, η(r) = 1 if |r| ≥ 2, and ηǫ (r) = η(r/ǫ). ′ ′ Set also φǫ (x′ , x3 ) := ηǫ (x3 )φ(x′ , x3 ), φ+ ǫ (x , x3 ) :=φǫ (x)χ(0,+∞) (x , x3 ), ′ − ′ ψǫ− (x′ , x3 ) :=φǫ (x′ , x3 )χ(0,+∞) (x3 ) and φ− ǫ (x , x3 ) :=ψǫ (x , −x3 ). 2 ∞ ± Then, φǫ ∈ C0 (R × (0, +∞)) and so R3 ± ∇u · ∇φ± ǫ − g(u)φǫ dx = 0 . As a consequence, R3 ∇U · ∇φǫ − f (U )φǫ dx ∇u(x′ , x3 ) · ∇φǫ (x′ , x3 ) − g(u(x′ , x3 ))φǫ (x′ , x3 ) dx + = R2 ×(0,+∞) ∂x3 u(x′ , −x3 )∂x3 φǫ (x′ , x3 ) − ∂x′ u(x′ , −x3 ) · ∂x′ φǫ (x′ , x3 ) R2 ×(−∞,0) +g(u(x′ , −x3 ))φǫ (x′ , x3 ) dx ′ ′ + ′ ∇u(x′ , x3 ) · ∇φ+ ǫ (x , x3 ) − g(u(x , x3 ))φǫ (x , x3 ) dx = R3 − R3 = 0.

Differential Equations, 51(1):126–150, 1984. K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. , 138(3-4):219–240, 1977. Enrico Valdinoci, Berardino Sciunzi, and Vasile Ovidiu Savin. Flat level set regularity of p-Laplace phase transitions. Mem. Amer. Math. , 182(858):vi+144, 2006.

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