Mathematics / advanced
Elliptic partial differential equations
Develop existence, uniqueness, and regularity for boundary-value problems using energy and comparison methods.
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Inside this module
Module 01 advanced
Elliptic partial differential equations
Develop existence, uniqueness, and regularity for boundary-value problems using energy and comparison methods.
- 01
Uniform ellipticity and principal symbols
Read the highest-order part of an operator and distinguish pointwise positivity from a uniform directional bound.
30 min - 02
Lax–Milgram and weak Dirichlet problems
Construct a weak solution from a bounded coercive bilinear form without assuming classical second derivatives.
32 min - 03
Coercivity and energy estimates
Track the constants that control a weak solution, including lower-order terms and approach to resonance.
32 min - 04
Maximum principles and comparison
Control solutions by their boundary and by barriers while tracking the operator sign and zero-order assumptions.
32 min - 05
Green's functions and Poisson kernels
Represent source and boundary effects with kernels whose normalization, signs, and domains are explicit.
33 min - 06
Interior regularity by difference quotients
Earn a second weak derivative through uniform translation estimates, and see how a coefficient jump can prevent it.
34 min - 07
Schauder estimates and Hölder data
Connect Hölder coefficients and forcing to controlled classical derivatives, with separate interior and boundary hypotheses.
34 min - 08
Calderón–Zygmund estimates
Control second derivatives in Lᵖ through singular-integral cancellation, with endpoint and coefficient limitations stated.
34 min - 09
Harnack inequality and positive solutions
Compare interior values of a nonnegative homogeneous solution and identify the roles of positivity, scale, and source terms.
33 min - 10
Fredholm boundary problems and solvability
Use a compact resolvent to identify finite-dimensional compatibility conditions and the source of nonuniqueness.
35 min