Interior regularity with coefficients on the domain #
Evans, Partial Differential Equations (2nd ed.), §6.3.1 poses its interior regularity theorems
on a bounded open U ⊂ ℝⁿ for coefficients given on U alone, with no bound on a^{ij} over
U, and for u ∈ H¹(U) with no boundary condition. The local chain of this development
(higher_interior_regularity_W12) runs on a global FullEllipticOp. This file reads the local
chain back on plain functions on U.
For an open V with closure V compact in U, exists_localOp_C1 or exists_localOp_Ck gives an
open W between closure V and U and a global operator agreeing with the given coefficients on
W. The solution restricted to W is a local weak solution of that operator in W12 W
(isLocalWeakSolution_of_localWeakSol), and the local chain on W at the compact closure V
gives the family on V (exists_family_of_localWeakSol). The constant is fixed by V, U and
the coefficients before the solution and the datum are, as in Evans.
Main declarations #
exists_family_of_localWeakSol: the transfer from a global operator agreeing with the coefficients nearclosure V.interior_H2_regularity_evans: Theorem 1 with the unnecessary boundedness and symmetry assumptions omitted.higher_interior_regularity_evans: Theorem 2 with the unnecessary boundedness and symmetry assumptions omitted.
Extending the class of g on U by zero and restricting to V ⊆ U gives the class of g
on V.
The L² norm over a smaller set is smaller.
A function in L^∞(U) is essentially bounded on U.
Transfer of the local chain to plain functions on U. Let Op agree with a, b, c on an
open W ⊆ U (everywhere for a, almost everywhere for b and c) and satisfy the order-k
conclusion of the local chain on W. For a compact K ⊆ W and a measurable V ⊆ K there is a
constant such that every u with weak gradient G, both in L²(U), solving the local weak
formulation on U with a datum f whose family of k weak derivatives on U is bounded by
M, has weak derivatives up to order k + 2 on V, bounded by C (M + ‖u‖_{L²(U)}).
In dimension zero every class has a family of every order, the family being constant, since there is no direction to differentiate in.
Equations
Instances For
The coefficients of Theorem 1 as a C1OpOn, with one essential bound for all transport
components.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Interior H²-regularity (Evans, Partial Differential Equations (2nd ed.), §6.3.1,
Theorem 1, p. 327), with the unnecessary boundedness and symmetry assumptions omitted. U ⊆ ℝᵈ is
open,
a^{ij} ∈ C¹(U), b^i, c ∈ L^∞(U), the operator is uniformly elliptic with a constant θ > 0
for almost every x ∈ U (§6.1.1, (4)). For each open V ⊂⊂ U there is
a constant C, depending on V, U and the coefficients alone, such that every
u ∈ H¹(U), given as u and its weak gradient G in L²(U), that is a weak solution of
L u = f in U for f ∈ L²(U) has weak derivatives up to order two in L²(V), which is
u ∈ H²(V), with
‖u‖_{H²(V)} ≤ C (‖f‖_{L²(U)} + ‖u‖_{L²(U)}).
Since every V ⊂⊂ U is covered, this is also u ∈ H²_loc(U). The norm is iteratedNorm, the
sum over lists of directions. The weak formulation is tested against C_c^∞(U), as in Remark
(ii) after the theorem; a solution tested against H₀¹(U) is one. The boundedness of U and
symmetry of a^{ij} assumed by Evans are unnecessary here.
Higher interior regularity (Evans, Partial Differential Equations (2nd ed.), §6.3.1,
Theorem 2, p. 332), with the unnecessary boundedness and symmetry assumptions omitted. Let m be
a nonnegative integer, U ⊆ ℝᵈ
open, a^{ij}, b^i, c ∈ C^{m+1}(U), the operator uniformly elliptic with a constant
θ > 0 for almost every x ∈ U (§6.1.1, (4)). For each open V ⊂⊂ U
there is a constant C, depending on m, U, V and the coefficients alone, such that every
u ∈ H¹(U) that is a weak solution of L u = f in U for f ∈ H^m(U) has weak derivatives up
to order m + 2 in L²(V), which is u ∈ H^{m+2}(V), with
‖u‖_{H^{m+2}(V)} ≤ C (‖f‖_{H^m(U)} + ‖u‖_{L²(U)}).
Since every V ⊂⊂ U is covered, this is also u ∈ H^{m+2}_loc(U). f ∈ H^m(U) is f ∈ L²(U)
with a family Hf of weak derivatives up to order m in L²(U), and both norms are
iteratedNorm. The weak formulation is tested against C_c^∞(U). The boundedness of U and
symmetry of a^{ij} assumed by Evans are unnecessary here.