Solving linear partial differential equations with constant coefficients

8.1.3.1 Fundamental solutions

Among the solutions to LPDE’s, some are particularly studied: these are the fundamental solutions, that is solutions Eto the equation:

DE“δ0

where the parameter is a Dirac distribution. Because partial differential linear operators with constant coefficients behave particularly well with respect to convolution, the answer to this particular input is enough to compute the answers to any inputf.

Definition 8.1.11. A fundamental solution for the LPDODconsists in a distributionψPD¹pRⁿq¹to the equation:

Dψ“δ0.

Remark8.1.12. Notice that such a fundamental solution is in general not unique: ifφis such thatDφ“0, then ED`φis also a fundamental solution toD.

Example8.1.13. Because of linear partial differential operators, we are working with distributions whose support is not necessarily compact. Indeed, the existence of a fundamental solution is not ensured when distributions must apply to any smooth function. The typical example is

D:f PC⁸pR,Rq ÞÑf¹,

wheref¹is the derivative of the real, one-variable functionf. Iff has compact support, one can define:

E_D:f ÞÑ ż0

8

f

and one has indeedDEDpfq “fp0q. This however is not possible in full generality whenf PC⁸pRⁿq. Observe that it would be enough for the partial derivative off to be bounded by an integrable function forxbig enough.

Then the derivative of f would be integrable, and ED could be defined. Extended to any partial differential operator, this says that if any partial derivativeB^αf,αPNⁿ off are integrable onRⁿ, thenEDcan be defined.

This idea is implemented in the construction of a tamed fundamental solutionE_DPS¹pR^Rq[43].

Notation 8.1.14. ConsiderD : EpRⁿq //EpRⁿqa LPDO. When a fundamental solution forD exists and it is fixed, it is denoted byE_DPD¹pRⁿq.

Remark8.1.15. We will recall later in theorem8.1.18that whenDis a LPDO with constant coefficients there is always of at least one fundamental solution forD.

The resolution of LPDOs with constant coefficients is always possible, and particularly elegant, due to the behaviour of convolution with respect to partial differentiation.

Proposition 8.1.16. ConsiderD “ ř

αaαB^α a LPDO with constant coefficients. Suppose that D admits a fundamental solutionED. Then for anyφPE¹pRⁿqwe have:

DpED˚φq “φ. (8.3)

Proof. Thanks to the bilinearity of the convolution product DpED˚φq “ÿ

α

aαrB^αpED˚φqs

“ pÿ

α

aαB^αEDq ˚φthanks to equation8.1and the bilinearity of˚,

“δ₀˚φby definition ofE_D,

“φ

Remark8.1.17. Beware that equation8.3is valid inD¹pRⁿq, but not inE¹pRⁿq. Indeed, the convolution between φPE¹pRⁿqandψPD¹pRⁿqonly yields a distributionψ˚φPD¹pRⁿq. This is by construction of the convolution following proposition7.3.16. It is also a particular case of the theorem of support [42, Thm 4.2.4] which says that the support of convolutionφ˚ψof two distributionsφPD¹pRⁿqandψPE¹pRⁿqis contained is the sum of the respective supports ofψandφ. We do not emphasize on support and singularities of distributions here.

Theorem 8.1.18. Every LPDEcc admits a fundamental solutionE_DPD¹pRⁿq.

1note that we do not ask forψPE¹pRⁿq, as this is not possible in general, even for the LPDO with constant coefficients

Comments on the proof. Several constructions of a fundamental solution exist: we refer to the one of Mal-grange, refined and exposed by Hormander [41, Thm 3.1.1] or to [42, 7.3.10] which gives a fundamental solution with optimal local growth. Others construct a fundamental solution which is temperate [43]. The proof starts with a technical Lemma [41, 3.1.1], which says that for any smooth function withcompact supportf P DpRⁿq, we have that

|fp0q| ďCkpcoshp|x|qDpfqk_1, 1

degP (8.4)

whereCdoes not depend off. Thus ifDpfq “ Dpgq, one hasDpf ´gq “ 0andfp0q “gp0q. Then defines the functionEˇ onDpDpRⁿqqas EpDpfˇ qq “ fp0q. This function is well defined, linear and continuous. The above majoration allows then to extend this function toEPD¹pRⁿqthrough Hahn-Banach3.3.6. The theorem is in fact much more precise as we have information about the local growth ofE_D. We do not have in general that E_DPE¹pRⁿq.

Remark8.1.19. Notice that equation8.4ensures thatE_Dis well defined onDpDpRⁿqq, as forf PDpDpRⁿqqthe functiongsuch thatf “Dgis unique. This is not true for functions inDpEpRⁿqq

8.1.3.2 The space!D

This section studies the question of which space is the good interpretation of!DRⁿfor " the space of distributions which are solutions to a LPDEcc "

Let us sum up the situation: the computationE_Dpgqis well defined as soon asgPDpRⁿq, as Theorem8.1.18 shows thatE_DPD¹pRⁿq. Iff PDpRⁿq, then obviouslyDpfˇ q PDpRⁿqand we have:

EDpDpfˇ qq “DEDpfq “fp0q.

Iff PEpRⁿq(that is,fdoes not necessarily have compact support) is such thatˇpDpfqq PDpRⁿq, thenE_DpDpfˇ qq is well defined, and we have alsoE_DpDpfˇ qq “fp0q. However, whenf PEpRⁿqandDpfˇ qdoes not have compact support, thenE_DpDpfˇ qqisnot defined, and thus we can’t computefp0qasE_Dapplied toDpfˇ q. Let us export this this reasoning to distributions.

Definition 8.1.20. We denote by!_DRⁿthe sub-vector space ofD¹pRⁿqconsisting of the distributionsφPD¹pRⁿq such thatDφPE¹pRⁿq.

!_DRⁿ :“ tφPD¹pRⁿq |DφPE¹pRⁿqu

Example8.1.21. A typical example of distribution in!_DRⁿisE_D, asDE_D“δ₀PE¹pRⁿq.

Proposition 8.1.22. Endowed with the topology inherited fromD¹pRⁿq, the space!_DRⁿ is a lcs. Moreover, we have the topological embeddings²E¹pRⁿqãÑ!DRⁿãÑD¹pRⁿq.

Proof. If we considerDas linear continuous endomorphism:

D:D¹pRⁿq //D¹pRⁿq

then!_DRⁿ is the inverse image ofE¹pRⁿqbyD. AsEpRⁿqis closed inDpRⁿq, and asD is continuous, then

!_D is a closed sub-locally convex topological vector space of D¹pRⁿq. It is Hausdorff: considerφ ‰ φ¹ are both distributions, and ifDφandDφ¹both have compact support. ConsiderV andV¹ disjoint neighbourhoods D¹pRⁿqofφandφ¹ respectively. Then asE¹pRⁿqis dense inD¹pRⁿq,V XE¹pRⁿqandV XE¹pRⁿqare two non-empty open sets contained in!DRⁿ whose intersection is empty and containing respectivelyφandφ¹. The topological embeddings follows from the fact that ifφhas compact support, thenDφhas compact support: thus E¹pRⁿq Ă !DRⁿ. This embedding is topological asD¹pRⁿqandE¹pRⁿqcarry the same topology. The other embedding follows from the fact that by definition,!DRⁿis a sub-lcs ofD¹pRⁿq.

Notation 8.1.23. We denote by?_DRⁿthe lcsp!_DRⁿq¹endowed with the strong topology on the dual.

AsDcommutes with convolution (proposition8.1.10) and as the convolution of two distributions with compact support has compact support, we have immediately:

Corollary 8.1.24. For anyφP!DRⁿand anyψPEpRⁿqwe haveφ˚ψP!DRⁿ.

Corollary 8.1.25. Forf P?_DRⁿandgPC⁸pRⁿq,Dpf˚gqis defined asDf˚g. It is coherent with the notation whenf PDpRⁿq, as partial differentiation commutes with convolution (Proposition8.1.10).

2that is, the linear continuous injections

Analogy with DiLL Defining!D₀Rⁿas the inverse image ofE¹pRⁿqbyD0inD¹pRⁿqdoes not make any sense.

Indeed, D0f is a linear function: it has compact support if and only if it is null. Thus with the definition used before, we have?D₀Rⁿ “ t0uand!D₀Rⁿ“Rⁿ.

Remark8.1.26. AlthoughD0is not a LPDOCC,δ0kind of behaves as a fundamental solution toD0. Indeed, if a functionf PC⁸pRⁿ,Rqis such that there isgPC⁸pRⁿ,Rqsuch that:

D0g“f, thenfis necessarily linear, and thus one can takeg“f “δ0˚f.

The good definition for “the space of distributions solutions toD0ψ“φ“ is

?_D0Rⁿ:“D^´1₀ pEpRⁿqq, and thus

!_D0Rⁿ»LpRⁿ,Rq¹ »Rⁿ. In that case dereliction and codereliction rules are interpreted in NUCLby:

dD₀ : !Rⁿ //!D₀Rⁿ, ψÞÑ p`ÞÑψp`qq d¯D₀ : !D₀Rⁿ //!Rⁿ, φ“evxÞÑ pf ÞÑφpD0pfqq.

However if we look for a fundamental solutionED₀PE², it must satisfy:

ED₀pD0fq “fp0q

and thus in particular, for any ` P E<sup>1</sup>, we haveED0p`q “ ED0pD0`q “ `p0q “ 0. Thus the analogy with fundamental solutions does not hold betweenD₀ and LPDOcc, outside of the interpretation of dereliction and co-dereliction.

Outlook 16. One should explore the similarities between the work of Ehrhard on anti-derivatives [20], and the proof theory of fundamental solutions as described in this chapter.