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Let B(0,1) = {(x,y) ∈ ℝ2|x2 + y2 < 1} be the open unit disc in ℝ2, ∂B(0, 1) denote the boundary of B(0,1), and v denote unit outward normal to ∂B(0, 1). Let f : ℝ2 → ℝ be a given continuous function. The Euler-Lagrange equation of the minimization problem  

\(\rm min \left\{\frac{1}{2}\iint_{B(0,1)}|\nabla u|^2dxdy+\frac{1}{2}\iint_{B(0, 1)}e^{u^2}dxdy+∈t_{\partial B(0, 1)}fuds\right\}\)

subject to u ∈ C1 \(\rm \overline{B(0, 1)}\) is

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  1. \(\rm \left\{\begin{matrix}\Delta u=-ue^{u^2}&\rm in \ B(0, 1)\\\ \frac{\partial u}{\partial \nu}=f&\rm on\ \partial B(0, 1\end{matrix}\right.\)
  2. \(\rm \left\{\begin{matrix}\Delta u=ue^{u^2}+f&\rm in \ B(0, 1)\\\ u=0&\rm on\ \partial B(0, 1\end{matrix}\right.\)
  3. \(\rm \left\{\begin{matrix}\Delta u=ue^{u^2}&\rm in \ B(0, 1)\\\ \frac{\partial u}{\partial \nu}=-f&\rm on\ \partial B(0, 1\end{matrix}\right.\)
  4. \(\rm \left\{\begin{matrix}\Delta u=ue^{u^2}&\rm in \ B(0, 1)\\\ \frac{\partial u}{\partial \nu}+u=f&\rm on\ \partial B(0, 1\end{matrix}\right.\)

Answer (Detailed Solution Below)

Option 3 : \(\rm \left\{\begin{matrix}\Delta u=ue^{u^2}&\rm in \ B(0, 1)\\\ \frac{\partial u}{\partial \nu}=-f&\rm on\ \partial B(0, 1\end{matrix}\right.\)
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The Correct answer is (3).

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