Partial differential equations: modern theory

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 46

ستتعلّم

• Work with distributions, weak derivatives and the Sobolev spaces H¹ and H¹₀.
• Prove existence and uniqueness of weak solutions of elliptic boundary value problems with the Lax–Milgram theorem.
• Use energy methods and the maximum principle for the heat and wave equations, and explain regularity of weak solutions.

تعريف

• A distribution on an open set Ω is a continuous linear functional on the test functions C_c^∞(Ω).
• The Sobolev space H¹(Ω) consists of L² functions whose weak first derivatives are in L².
• A weak solution of −Δu = f in Ω with u = 0 on ∂Ω is a u ∈ H¹₀(Ω) with ∫ ∇u · ∇v = ∫ f v for all v ∈ H¹₀(Ω).

مبرهنة

• Lax–Milgram: a bounded coercive bilinear form on a Hilbert space represents every bounded linear functional uniquely.
• Poincaré inequality: for bounded Ω, ‖u‖{L²} ≤ C ‖∇u‖{L²} for u ∈ H¹₀(Ω).
• Weak maximum principle: a harmonic function on a bounded domain attains its maximum on the boundary.
• Interior elliptic regularity: weak solutions of −Δu = f with smooth f are smooth in the interior.

مفاهيم خاطئة شائعة

• Thinking every PDE has classical solutions; many natural problems only have weak solutions, which may later be shown to be smooth.
• Believing well-posedness is automatic; Hadamard's example shows the Cauchy problem for Laplace's equation depends discontinuously on the data.

يقود إلى

اكتشافات اختيارية

• Geometric analysis Geometric analysis uses nonlinear PDE (minimal surface equation, harmonic maps, Ricci flow) to answer questions in geometry and topology.
• Numerical analysis The finite element method is the Galerkin method for weak formulations: Lax–Milgram and Céa's lemma explain why it converges.

مسارات التعلّم

• Analysis, PDE and dynamics

قراءات إضافية

• Lawrence C. Evans, Partial Differential Equations (1998)
• Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011)